v.,..oo, /31 1
Variational S1-symmetric problems with resonance at infinity
139
To compute D EG Zjio (Vi, D", (V)) it is enough to compute degree of a linear isomorphism V2i(0)1V~ ' To compute DEG zjio {Vi, D/31 (V{:JO) x D/32(V200 )) it is enough to compute degree of a linear isomorphism V 2 f (oo) 1V OO and degree 2 of Vcpoo. Applying Corollary 4.3 of [12) we obtain
DEG z "0. (Vf,D",{V)) = (3.17)
2'1 . DEG S
0
I
.
(Vcpo, D"'(V1 )) . (-I)
m-(,,7 2/(0)IV O) 2
•
m-
(2 ) V f{O)IR[k;o.i;ol
and
DEGz jio {Vi, D/31 (V100 ) { -1 )
m- (V2/(OO)IVOO) 2
•
X
D/32 (V2OO))
=
(3.18)
2'1 . m _ ( V 2 f (00) IIR[kio ,j;ol ) +
{_I)m- (V2 f(OO)1V2OO ) . DEGz .. (Vcpoo, D/31 (V100 )). "0
By assumption 2. and by Lemma 2.16 of [13) we have
DEGz"0 ·. {Vcpoo, D/31 (V100 ))
= ±1,0.
By (3.16), (3.17), (3.18) and assumption 4. we obtain DEGzj;o (V f, 0) '" O. From Theorem 3.9{a) of [12) it follows that V f- 1 (O) n OZj;o tion.
'"
0, a contradic-
o Theorem 3.7 is dual to Theorem 3.5. We have exchanged in these theorems assumptions at the origin with the assumptions at infinity. Theorem 3.7 Let Vi E OSI
1.}l
A.c~{V).
= {O} and V1°O n
SI
Assume that
= {O}, nl
2. V ~ EBIR[ki,ji), V10 ~ IR[I,jio)' V100 ~ EBIR[kt,jlJ, i=1
i=1
3. jio '" gcdof {a1, ... , ak} for any a1, . . . , ak E {it, · . . , j;1 },
4·
for'"Y
= ±2,0.
N. Hirano and S. Rybicki
140
Proof. Suppose contrary to our claim that there are no nontrivial zeros of \1 j i.e. \1 j-l(O) = {O}. Repeating the first part of the proof of Theorem 3.2 we obtain positive numbers a., (31, (32 such that formulas (3.2), (3.3) hold true. Put D = (Dfh (Vt') x D.B2(V2OO )) - cl(D",(V)) . By Theorem 3.9.(c) of [12]) we obtain
DEG(\1 j, D) = DEG(\1 j, D.Bl (V1OO) x D.B2(V2OO )) -DEG(\1 j, D",(V)). (3 .19) What is left is to show that DEG(\1 j, D) is a nontrivial element in U(SI). More precisely, we will show that
is a nonzero integer. By assumption 1. and by the definition of degree for Sl--equivariant gradient maps we have
By assumption 3. it follows that the isotropy group of any point in V1°o is different from Zjio. Therefore, by the definition of degree for Sl--equivariant gradient maps we obtain DEG z . (\1cpoo, D"'(Vl00 )) = O. Summing up, we ob"0 tain the following
In order to compute D EG z .. (\1 j, D", (V)) it is enough to compute degree of "0 a linear isomorphism \1 2 j(0)W20 and degree of \1cpo. Moreover, to compute DEG zjio (\1 j, D.Bl (V1OO) x D.B2(V200 )) it is enough to compute degree of a linear isomorphism \1 2j (00) W2 Applying Corollary 4.3 of [12] we obtain °O.
DEG ZjiO (\1j,D",(V)) (-1)
m-
(V2/(O)IVO) 2
1
(2
· 2· m- \1 j(O)IR[kio,jiol
)
=
(3.20)
+
DEGzjio (\1cpo, D"'(VI0))) DEG zjio (\1 j, D.Bl (Vl00 )
X
D.B2 (V2OO))
_ ( 2 ( ) ?1 (-1 )m- (V2/(oo)IVOO) 2 • m \1 j oo)IR[kio.iiol) .
(3.21)
Variational SI-symmetric problems with resonance at infinity
141
By assumption 2. and Lemma 2.16 of [13] we have DEGz ·. (V'ipo, D{31 (Vt' )) = "0 ±1, O. By (3.19) , (3.20), (3.21) and assumption 4. we obtain DEGz . (V' f, 0,) i:O. From Theorem 3.9(a) of [12] it follows that V' f- 1(0) n O,z;;o diction.
"0
i:- 0, a contra-
o Theorem 3.8 corresponds to Theorem 3.6. We have exchanged assumptions at the origin with assumptions at infinity. In Theorem 3.8 we do not consider the case k = 2 because we know nothing about the Brouwer degree of a twodimensional gradient map computed on a disc of a large radius. Theorem 3.8 Let V' f E
1. ~o
81
= {O}
A.ci'" (V) . Assume that
and dim Vt'
81
= k,
n
2. V ~
EB lR[ki ,ji], V
10
n1
~ 1R[I,jio]' vt' ~ lR[k,O] E9
i =1
3. jio
EB lR[k;'jfl, i =1
i:- gcdof {aI, .. . , ak}
4· m- (V'2f(0)I1R[k;0,j;01)
for any aI, ... ,ak E {it, ... ,j;l },
+1'2
i:-'Yl· m -
(V' 2f(OO)I1R[k;0 ,j ;01) , where
(a) if k = 1, then 1'1 = ±1, 0 and 1'2 = ±2, 0, (b) if k > 1, then 1'1 E Z and 1'2 = ±2,0.
Proof. Suppose contrary to our claim that there is no nontrivial zeros of V' f i.e. V' f- 1 (0) = {O} . Repeating the first part of the proof of Theorem 3.2 we obtain positive numbers a, (31, (32 such that formulas (3.2), (3.3) hold true. Put 0, = D{31 (Vt') x D{32(V200 ) - cl(D",(V)) . By Theorem 3.9.(c) of [12]) we obtain
DEG(V' f, 0,)
= DEG(V' f, D{31 (Vl
00
)
XD{32(~OO)) -DEG(V' f, D",(V)). (3 .22)
What is left is to show that DEG(V' f, 0,) is a nontrivial element in U(SI) . More precisely, we will show that
is a nonzero integer. By assumption 1. and by the definition of degree for SI--equivariant gradient maps we have
N. Hirano and 8. Rybicki
142
From the assumption 3. it follows that the isotropy group of any point in Vt' is different from Zjio. Therefore, by the definition of degree for 8 1 -equivariant ·. (\1£{Joo, D{31 (V100» = O. Summing up, we gradient maps we obtain DEGz"0 obtain the following
To compute DEGz .. (''ii'I, Da(V» it is enough to compute degree of a linear "0 isomorphism \12/(0)JV20 and degree of two-dimensional map \1£{Jo. Moreover, compute DEGz ·. (\11, D{31 (V100 ) x D,62(V2OO it is enough to compute degree "n of a linear isomorphism \1 2/(oo)JV2 and DEG 81 (\1£{Joo, D,61 (V1OO». Applying Corollary 4.3 of [12] we obtain
»
OO
DEGZiiO (\11, Da(V» 1 m- (V 2'. (-1)
2
!(O)JV O ) 2
•
m-
(2 ) \1 I(O)pR[kio,jiol
(3.23)
+
DEG z ·. (\1£{Jo,D a (l/iO» "0 and
DEGziin (\11, D,61 (V1OO) x D,62(V2OO »
=
(3.24)
~ . DEG 8 1(\1£{Joo, D,61 (vt'» (_l)m- (V2 !(oo)JV2oo ) 2
m- (\1 /(oo)PR[k io,jiol) .
By assumption 2. and Lemma 2.16 of [13] we have DEG z ·. (\1£{Jo, D a (V10» = "0 ±1, O. By (3.22), (3.23), (3.24) and assumption 4. we obtain DEGz .. (\11, n) '" "0 O. From Theorem 3.9(a) of [12] it follows that \11- 1(0) n nZjio '" 0, a contradiction.
o The following theorem gives us information about nontrivial zeros of asymptotically linear 8 1 -equivariant gradient maps in the case when the kernels of the linearizations at infinity and at the origin are two-dimensional nontrivial representations of the group 8 1 of the same type.
Variational SI-symmetric problems with resonance at infinity
143
Theorem 3.9 Let \1/ E A.c;x>(V) . Assume that n
1. V ~ EBlR[ki,jd, V10 ~ lR[l,jio]' Vl°O ~ lR[l,j io]' i=1
2.
where 'Y
= ±4, ±2, O.
Proof. Suppose contrary to our claim that there is no nontrivial zeros of \1/ i.e. \1/- 1 (0) = {O}. Repeating the first part of the proof of Theorem 3.2 we obtain positive numbers 0.,(31 , /32 such that formulas (3.2), (3.3) hold true. Put 0= (D/31 (V1OO) XD/32 (V200 )) -cl(Da(V)). By Theorem 3.9.(c) of[12]) we obtain
What is left is to show that DEG(\1 /,0) is a nontrivial element in U(SI). More precisely, we will show that
is a nonzero integer. By assumption 1., Lemma 2.16 of [13] and by definition of degree for SI--equivariant gradient maps we have
DEGSI (\1 CPo , D a (Vlo)) = 1, DEGz ·. (\1cpo, Da(~O)) = ±1,0,
(3.26) (3.27)
"0
DEG S1(\1cpoo, D/31 (Vl00 )) = 1, DEGz .. (\1cpoo, D/31 (VlOO)) = ±1 , O.
(3.28) (3.29)
"0
By (3.26) and (3.28), we obtain
DEGz .. (\1/, Da(V)) = DEGz; (\1 2 /(0) , Da(V20))+ "0 '0 DEGSI (\1 2 /(0), D a (V20)) . DEGz;,o (\1cpo, Da (VIO)), 2 00 DEGz;.o (\1/, D/31 (VlOO) x D/32(V200 )) = DEGz;.o (\1 /(00), D/32(V2 ))+ DEG SI (\1 2/(00), D/32(V2OO )) . DEGz;.o (\1cpoo, D/31 (~OO)) .
N. Hirano and S. Rybicki
144
Applying Corollary 4.3 of (12) we obtain
=
(3.30)
) + 2'1 ·m- (2 V1 j(O)I1R[k' o ,j'ol
(3.31)
·. (V1 f, Da(V» DEG z "0 (-1)
m-
('\7 2f(O)IVO ) 2
•
DEGz . (V1cpo, D a (VI0))), "0
DEGz i • (V1 f, D/31 (vtO) x D/32 (V2OO» o
( -1 )
m-
=
(3.32)
('\72 f(oo)lVoo) 1 - (M2f() ) + 2 • 2' . m v 00 IIR[k'o,j'ol DEGz"0 ·. (V1CPoo, D/31 (VlOO))).
By assumption 2., DEG z .. (V1f,n) :f
(3.25), (3.27), (3.29), (3.30), (3.32) we obtain O. From Theorem 3.9(a) of (12) it follows that
"0
V1 f- 1 (0)
n nZi.o :f 0, a contradiction.
o n
Remarks 3.10 Let V ~
EB lR[k
i,
ji) be representation of the group SI consid-
i=1
ered in theorems of this section. Assume additionally that 1. {j E {iI, ... ,jn} : .,L E N} J,o
2. V1 f- 1 (0) n Vs
1
= {jio}'
= {O} .
Then isotropy group of a nontrivial solution, whose existence was proved in theorems of this section, is equal to 'L j •o • Remarks 3.11 Let V1 f E A£OO(V) and assume that V1 2 f(oo) is degenerate.
Applying Splitting lemma at infinity and Cartesian product formula we can sometimes show that DEG(V1f,D/3(V» is nontrivial in U(SI), where f3 » O. Therefore, V1 f- 1 (0) n D/3(V) :f 0. At the end of this section we present example which illustrates how the machinery presented in this section works. Namely, we show an example of a map such that all the assumptions of Theorem 3.2 are fulfilled for this map and that the classical topological invariants are not applicable in this situation. Example 3.12 Let V1 f E
V12f(0) =
A£~(V)
[I~2 ~],
and V
= lR[l, l)9lR[l, 2).
V1 2f(00) =
Let us verify assumptions of Theorem 3.2
Assume that
[-~d2 ~].
Variational S1-symmetric problems with resonance at infinity
2. jio
145
= 1,
(a) jio ::j:. gcdof {2},
(b) jio ::j:. gcdof {2}, (c) O::j:.
t · 2.
Applying Theorem 3.2 we prove the existence of nontrivial solution of the equation 'V f(v) = O. On the other hand notice that it is impossible to apply to our problem results presented [2J, [3J and!4J. Namely, the above mentioned results ' do not work if we restrict our problem to the set of fixed points of the action of any subgroup of S1 .
In the following theorem we formulate sufficient conditions for the continuation of critical points of a family of S1-equivariant gradient maps.
Theorem 3.13 Consider a family f>.. : V -+ IR[l, 0], A E IR of S1-equivariant C 2 -maps . Assume that there is Ao E IR such that 'V fAo satisfies assumptions of one of theorems of this section. Then one of the following possibilities hols: 1. 0 E V is not isolated in 'V f~1(0), 2. for any M
3. there are
>0
Goo
there is
VM E
'Vf~1(0) ' such that
IlvM11 > M,
> Go > 0 such that there exist continua C- ,C+ with C- c V x (-00,Ao]n'Vf- 1(0), C+ c V x [Ao, +00) n 'V f- 1 (0),
and for both C = C- , C = C+ the following statements are valid: (a) C n DO:60 (V) - cl(Do:o(V)) x {Ao}) ::j:.
0,
(b) if V
S1
= {O},
(c) if V
S1
::j:. {O}, then either C is unbounded or else (0, AO) E C.
then C is unbounded,
Proof. Suppose that conditions 1. and 2. of thesis of this theorem are not satisfied. Repeating proof of one of theorems of the previous section we show that DEG('VfAo' Do: (V) - cl(Do:o(V))) ::j:. e E U(S1). Notice that if V S1 = {O} then 'V fA (0) = 0 for any A E IR. The rest of the proof of this theorem is a direct consequence of Theorem 5.1. 00
o
146
4
N. Hirano and S. Rybicki
Final Remarks
In this paper we have prepared a topological tool which we will apply to qualitative studies of solutions of elliptic differential equations, Hamiltonian systems and wave equations. We will study solutions of asymptotically linear differential equations with resonance at the origin and at infinity. Mainly we will be interested in the existence, bifurcations and continuation of solutions of such equations.
5
Appendix
In this appendix, for the conveniences of the reader, we present without proof theorem describing continuation of critical points of a family of Sl-equivariant Cl-maps. In fact in order to prove this theorem it is enough to repeat all the steps of the classical proof, replacing the Brouwer degree with the degree for Sl-equivariant gradient maps. Theorem 5.1 Let fA: V --+ 1R[I,O],A E IR. be a family of Sl-equivariant C 1 maps. Assume that there exist an open, bounded, S1-invariant subset n c V and AD E IR such that'9fAo1(O)nan = 0 and that DEG('9fAo' n) f:. e E U(SI) . Then there exists continua C- ,C+ with
C-
c V x (-00, AD] n'9r l (O),
C+
c
V x [AD, +00) n '9r l (O),
and for both C = C- and C = C+ the following statements are valid: 1. Cnn x {AD}
f:. 0,
2. either C is unbounded 01' else C n (V - cl(n)) x {AD}
f:. 0.
References [1] J. F. Adams, Lectures on Lie Groups, W. A. Benjamin Inc., New York Amsterdam, (1969),
[2] T . Bartsch and S. Li, Critical Point Theory for Asymptotically Quadratic Functionals and Applications to Problems with Resonance, Noni. Anal. TMA 28, No 3, (1997), 419-447,
Variational S1-symmetric problems with resonance at infinity
147
[3) K. C. Chang and J . Q. Liu, A Strong Resonance Problem, Chinese Ann. of Math. lIB, (1990), 191-210, [4) K. C. Chang, J. Q. Liu and M.J. Liu, Nontrivial Periodic Solutions for Strong Resonance Hamiltonian systems, Ann. lnst. Henri Poincare, Anal. Non. Lineaire 14, No 1, (1997) , 103-117, [5) E. N. Dancer, Degenerate Critical Points, Homotopy Indices and Morse Inequalities, J. fur Reine und Angew. Math. 350, (1984), 1069-1076, [6) E. N. Dancer, A New Degree for S1-invariant Gradient Mappings and Applications, Ann. lnst. Henri Poincare, Anal. Non Lineaire 2, No.5, (1985), 329-370, [7) E . N. Dancer & S. Rybicki, A Note on Periodic Solutions of Autonomous Hamiltonian Systems Emanating from Degenerate Stationary Solutions, Diff. and Int. Equat. 12, No.2, (1999), 1 -14, [8) T . tom Dieck, Transformation Groups, Walter de Gruyter, 1987, [9) K. G~ba, 1. Massab6 & A. Vignoli, On the Euler Characteristic of Equivariant Vector fields, Boll. U. Math. Italiana 4A, (1990), 243-251, [10) K. G~ba, Degree for Gradient Equivariant Maps and Equivariant Conley Index, Topological Nonlinear Analysis, Degree, Singularity and Variations, Eds.: M. Matzeu i A. Vignoli, PNLDE 27, Birkhauser, (1997), 247-272, [11) A. Maciejewski i S. Rybicki, Global Bifurcations of Periodic Solutions of Henon-Heiles System via Degree for S1-equivariant Orthogonal Maps , Rev. in Math. Ph. 10, No.8, (1988), 1125-1146, [12) S. Rybicki, A Degree for S1-equivariant Orthogonal Maps and Its Applications to Bifurcation Theory, Nonl. Anal. TMA 23, No 1, (1994),83-102, [13) S. Rybicki, Applications of Degree for S1 - equivariant Gradient Maps to Variational Nonlinear Problems with S1-symmetries, in Top. Meth. in Nonl. Anal. 9 , No.2, (1997), 383-417, [14) S. Rybicki, On Rabinowitz Alternative for the Laplace- Beltrami Operator on sn-1. Continua that Meet Infinity, Diff. and Int. Equat. 9, No.6, (1996), 1267-1277, [15) S. Rybicki, Global Bifurcations of Solutions of Elliptic Differential Equations, J. of Math. Anal. and Appl. 217, (1998), 115-128,
148
N. Hirano and S. Rybicki
[16] S. Rybicki, On Bifurcations from Infinity for Sl-equivariant Potential Operators, Nonl. Anal. TMA 31, No. 3/4, (1997), 343-361, [17] S. Rybicki, On Periodic Solutions of Autonomous Hamiltonian Systems via Degree for Sl-equivariant Gradient Maps, Nonl. Anal. TMA 34, No. 4, (1998) , 537-569, [18] S. Rybicki, Periodic Solutions of Vibrating Strings; Degree Theory Approach, accepted for publication in Ann. di Mat. Pura ed Appl., (1999) , [19] Z. Q. Wang, Equivariant Morse Theory for Isolated Critical Orbits and Its Applications to Nonlinear Problem, Lect. Notes in Math. 1306, Springer (1988), 202-221.
MUltiplicity results for Hamiltonian Systems with nonconvex Hamiltonian functions Marek Izydorek Abstract A certain class of Hamiltonian functions containing non--convex maps is considered. Multiplicity results for periodic solutions of corresponding Hamiltonian systems are derived. As a tool the Sl-equivariant cohomological Conley index in Hilbert spaces is used.
1
Introduction
In [15] an extension of the G-equivariant Conley index theory developed by
Floer (cf. [11]) to infinite dimensional spaces has been defined. Among the others, the cohomological version of the index was made. To every isolating G-neighbourhood X for a certain local G-.cS-flow 'fJ it assigns a graded cohomology group Ha(X) which admitts a H*(BG)-module structure, where BG stands for a classifying space of the group G. As it has been shown in [15] this new tool turned out to be very useful in studying certain strongly indefinite problems with symmetries. In particular, problems concerning multiplicity results for periodic solutions of autonomous Hamiltonian systems are of that type since such systems admit natural symmetries of the group 8 1 = {z E C, ; I z 1= I}. There are many papers concerning that subject. Let me only mention the following articles: [1],[3],[4],[5],[9],[10], [12],[20] and books: [2],[6],[16]. Given a Hamiltonian function Q E C 1 (JR 2 m , JR) consider the Hamiltonian system of differential equations ,.r·t i
= J'\lQ(z)
(1.1)
where J is the standard symplectic matrix and '\l denotes the gradient with respect to the inner product in JR 2 m. 0Research supported by Alexander von Humboldt Foundation and KBN grant 2-P03A00616
149
Marek Izydorek
150
Our aim. is to study multiplicity results for 27r-periodic solutions of (1.1) for a special class of Hamiltonian functions. Namely, we assume on Q to be of the form Q(z}
= F(II Bz 112}
where, F is a C l -map satisfying conditions (F.l}-(F.3) and B : ~2m -t ~2m is a linear isomorphism. This class contains non-convex Hamiltonian functions which appear when F(x 2 } is non-convex. For instance, if F(x} = e-'" then
is non-convex with a unique critical point at 0 E ~2m . Two cases will be discussed. First one, when lim"'-too F'(x} = 0, leads us to SI-equivariant asymptotically linear problem with resonance at the infinity. The second one, when lim"'-too F' (x) = ±oo, can be "approximated" by a sequence of asymptotically linear problems. In our considerations the G-equivariant cohomological Conley index defined in [15] will be used, with G = SI.
2
Equivariant Conley index and its cohomology
2.1
G-.L:S-flows.
Assume, we are given a linear and orthogonal action of a compact Lie group G on a Hilbert space H and a linear bounded G-equivariant operator L : H -t H such that the following conditions are satisfied:
• H = ffJ::'=oHn with all subspaces Hn being mutually orthogonal Grepresentations of finite dimension; • L(Ho} C Ho, Ho is the invariant subspace of L corresponding to the part of spectrum O'o(L) := i~ n O'(L) lying on the imaginary axis and L(Hn) = Hn for all n > 0; • O'o(L) is isolated in O'(L) i.e. O'o(L) n cl(O'(L) \ O'o(L))
= 0.
We say that f : H -t H is an CS -vector field if there exists a completely continuous and locally Lipschitz continuous map K : H -t H such that
f(x)
= Lx + K(x)
for all x E H.
Multiplicity results for Hamiltonian systems
151
If, in addition, f(gx) = gf(x) for x E Hand 9 E G then f is called a G-£S-
vector field. If f : H -t H is an £S-vector field and x E H, then there exists the maximal CI-curve (a(x), w(x)) :;) t I-t TJ(t, x) E H satisfying
dTJ dt { TJ(O,x)
fOTJ x
Moreover, if we set D(TJ) := ((t,x) E IR x H : a(x) < t < w(x)}, then D(TJ) C IR x H is open and TJ : D(TJ) :;) (t,x) I-t TJ(t,x) E H is continuous. If f is a G-£S-vector field then D(TJ) is a G-subset of IR x H, i.e. (t, x) E D(TJ) implies (t,gx) E D(TJ), and TJ(t,gx) = gTJ(t,x) for all (t,x) E D(TJ), 9 E G. In what follows we call TJ the local (G-)ftow generated by f. If TJ is a local G-flow and X is a G-subset of D(TJ) then
Inv(X, TJ)
:=
{x E X
j
TJ(t, x) E X for all t E 1R}
Inv(X, TJ) is the maximalTJ-invariant subset of X. Clearly, it is also a G-subset of X. We say that a bounded and closed G-subset X of H is an isolating Gneighbourhood for a local G-flow TJ iff X C D(TJ) and Inv(X) C int(X). The maximal TJ-invariant G-subset of X is included in the interior of X.
2.2
Homotopy G-.cS-index.
In order to define the homotopy Conley index we need a notion of G-spectra.
We let SV := VU{ 00 } to be a one-point-{;ompactification of a finite-dimensional real G-representation V with {oo} as base point. Let ~ = (Vn)g" be a fixed sequence of real G-representations of finite dimensions. Consider a sequence (En)~E) of compact, metrizable G-spaces with base points. Assume, we are given a sequence of G-maps (en: SVnEn -+ En+1)~E)' where SVn En := SVn 1\ En is the smash product. 2.1. Definition. We say that a pair E(~) := «En)~E)' (en)~E)) is a Gspectrum of type ~, if there exists no 2: n(E) such that en : SVn En -+ E n+ 1 is a G-homotopy equivalence for all n 2: no ·
Marek Izyd,
152
A G-map 01 spectra I : E(€) -+ E'(€) is a sequence of G-maps (In : E no 2: max{n(E),n(E')} such that diagrams
E~)~,
are G-homotopy commutative for all n 2: no· Two G-maps of spectra I, f' : E(€) -+ E'(€) are G-homotopic, if the] nl 2: 0 such that In is G-homotopic with I~ whenever n 2: nl· A G-map I : E(€) -+ E'(€) is said to be a G-homotopy equivalenc
G-spectra E(€) and E'(€), ifthere exists a G-map h: E'(€) -+ E(€) such
hoI is G-homotopic with the identity map
id E ({)
I
and 10 h is G-homot,
with idEI({). Two G-spectra E(€),E'(€) are G-homotopy equivalent, or they have same G-homotopy type, if there is a G-homotopy equivalence
I : E(€:
E'(€). We will denote [E(€)] for the G-homotopy type of a G-spectrum E(€) 2.2. Remark. For a given spectrum E = «En)~E)' (€n)~E») its G-homo1 type is uniquely determined by the G-homotopy type of a pointed G-SI
En, with n sufficiently large. In particular, if in the spectrum E the seqUl (€n)~E) is replaced by another sequence of G-homotopy equivalences (€n)~ then the resulting spectrum has the same G-homotopy type as the original, Therefore, in order to define a G-homotopy type [E(€)] one only needs a quence of G-spaces E = (En)~E) such that SVn En is G-homotopy equiva to En+! , for n sufficiently large. We are now in a position to define the equivariant homotopy Conley in, Following ideas of the non-symmetric case presented in [13] and [14] its, struction has been thoroughly described in [15]. Since [15] is recently submi we recall it briefly for our convenience. We keep all the assumptions and notations as before. Let Pn the orthogonal projection onto Hn :=
H -+ E EBi=oHi. Let H~ (resp. H;;), n ~ :
153
MUltiplicity results for Hamiltonian systems
denote the L-invariant subspace of Hn corresponding to the part of spectrum of L with the positive (resp. negative) real part. Since L commutes with all elements of G it follows that both spaces H: and H;; are representations of G. Define a sequence of G-representations: ~ = (Vn )8", Vn := H;;+l . Assume additionally, that f(x) = Lx+K(x) is a ~ubquadratic G-£S-vector field i.e., f satisfies \Ix E H, 1< K(x),x >I~ a II x 112 + b, where a and bare positive constants. Let TJ : IR x H -t H be the £S-flow generated by f and let X c H be an isolating G-neighbourhood for the flow TJ. Define fn : Hn -t Hn and Fn : Hn+I x [0, 1]-t Hn+I by
fn(x)
:=
Lx + Pn(K(x)) and Fn(x, t)
:=
Lx + (1 - t)Pn(K(x))
+ tPn+I (K(x)).
Let TJn : IR x Hn -t H n denote the G-flow induced by f n and ~n : IR x Hn+I x [0,1] -t Hn+I denote the family of G-flows induced by Fn. There is no EN such that Xn := X n Hn is an isolating G-neighbourhood for the flow TJn and for the family of flows ~n, whenever n ~ no. Choose n ~ no and set Sn := Inv(Xn, TJn). Thus, Sn admits a G-index pair (Yn , Zn) (see [11]) and the equivariant Conley index of Sn is the G-homotopy type of the pointed G-space
Yn/Zn . Let D! := {x E H"j: ;
II x
aD!
:=
II~ I}, D;; := {x E H;; ;
{x E H"j: ;
II x
II~ I},
II x 11= I}.
Let Sn+I,n := Inv(Xn+I x [0, 1], ~n), Sn+I,n(t) := {x E Xn+I ; (x, t) E Sn+I,n}. Clearly,
is a G-index pair for the isolated invariant set Sn = Sn+I,n(O) with respect to the flow ~n (-, ·,0) . Thus, the equivariant Conley index of Sn with respect to
~n(-'·, 0)
equals
the G-homotopy type of
(Yn x D~+I x D;;+I)/(Zn x D~+I x D;;+I U Yn
X
aD~+I x D;;+I)
which in turn is equal to the G-homotopy type of SVn (Yn/ Zn). Moreover Xn+I is a G-invariant isolating neighbourhood for the flow ~n(-'·, s) for all s E [0,1].
Marek Izydorek
154
Therefore, by the continuation property of the equivariant Conley index ( see [11)) SVn (Yn/ Zn) is G-homotopy equivalent with Yn+1 / Zn+l· Thus, in view of Remark 2.2 , the sequence (En)~ := (Yn/ Zn)~ determines uniquely the G- homotopy type [E(O) · 2.3. Definition. Let TJ be a G-.cS-flow generated by a sub quadratic G-.cSvector field and let X be an isolating G-neighbourhood for TJ· Define h~s(X, TJ) := [E(~)).
We call h~s(X, TJ) the G-equivariant .cS-homotopy Conley index of X with respect to TJ or simply, the G-.cS- index. Changing f outside an isolating G- neighbourhood X the general case can be reduced to a sub quadratic one (cf.[13)) .
2.3
Cohomology G-£S-index.
Let p : EG -t BG denote the universal principial G-bundle. To every G-space X one associates fibre bundle with fibre X, px : Xa
= EG
Xa X -t BG .
We apply the Alexander-Spanier cohomology to the bundle Px and for a pointed G-space (X, xo) we set: Ha(X) := H*(Xa, {xo}a) = H*(EG Xa X,EG Xa {xo}) . If A is aG-subspace of X , then Aa is a subspace of Xa . We set Ha(X, A) :=
H*(Xa, Aa). The fibre map px defines H * (BG) - module structure in Ha(X, A) j
for every a E H*(BG) and u E Ha(X , A) we let a * u := px(a) U u, where . U . : H*(EG Xa X) i8) Ha(X, A) -t Ha(X, A) is the cup- product (see [18)) . In particular, Ha(X)
= Ha(X, {xo})
is a H*(BG)-module. A G-map f :
(X,A) -t (Y,B) defines fa:= id Xa f: (Xa , Aa) -t (Ya,Ba) whose induced
homomorphism fa : Ha(Y, B) -t Ha(X, A) preserves H*(BG)-module strucis called the G-equivariant Borel cohomology (see ture. The functor H
a(·)
[8)). Let V be an orthogonal representation of G of dimension n, with unit ball (resp. unit sphere) DV (resp. SV). Assume, the action on V is orientation
Multiplicity results for Hamiltonian systems
155
preserving i.e. for each 9 E G the linear map corresponding to 9 is an element of GL+(V) . Then, the vector bundle ( : EG XG (V x X) -t EG XG X is orientable over integers. Using Thorn's isomorphism for (
T* : H*(EG
XG
X) -t H*+n(EG
XG
(DV x X), EG
XG
(SV x X»
one defines the suspension isomorphism of H*(BG)-modules
(cf. [15)) . 2.4. Remark. If V is a unitary representation of G and r(V) is the underlying real G-representation, then the action of G on r(V) is orientation preserving. On the other hand, each vector bundle is orientable over Z2.
In what follows, we assume ~ = (Vn)o to be a sequence of orthogonal Grepresentations such that, for each n the action of G preserves the orientation of Vn . Let E(~) = ((En)~E)' (cn)~E» be a G-spectrum of type ~ . Define a map p: NU {O} -t Nu {O}
p(O) = 0
p(n) = v(EB~~olVn) '
and
where v(V) is the dimension of V. For a fixed q E Z we are given a sequence of cohomology groups
and a sequence of isomorphisms
cq+p(n+1) . H q+p(n+1)(E
"n
.
G
n+l
) --+ H Q+p(n+1) (SVn E ) G
n
induced by G-homotopy equivalences Cn : SVn En -t E n+"l, n sufficiently large. Thus, for n big enough, the composition h~+p(n+1) in the following diagram
q+p(n+1) Hb+ p(n+1) (En+1) _ _ _c_n_ _ _ _ _~> Hb+ p(n+1) (SVn En)
.) "I
Hb+ p(n+l) (E +1 ) n is an isomorphism.
hQ+p(n+l) ____ G_ _ _ _ _ _>7 ~
1(5')-' Hb+p(n) (En)
Marek Izydorek
156
2.5. Definition. The q-th Borel cohomology group of a G-spectrum E E(~)
=
is the inverse limit group
H~(E) := lim{Hb+p(n) (En), hh+p(n)} . r
2.6. Remark. Directly from the definition one has Ha(E) ~ Hb+p(n) (En), for n large enough. It is also easily seen, that Ha(E) := ffiqEZH~(E) admits H*(BG)- module structure. A G-map of spectra f : E -+ F induces a homomorphism fa : Ha(F) -+ Ha(E) of H*(BG)-modules . Let f : H -+ H be G-£S-vector field . Let X be an isolating G-neighbourhood for the local flow 1} generated by f, and let Ex stands for corresponding Gspectrum. 2.7. Definition. We say that S
= Inv(X,1})
is homologically pt-hyperbolic
of index p E Z if there exists u E H~ ( Ex) , so that the map H*(BG)
-+ Ha(Ex) : a -+ a * u
is an isomorphism. The notion of homologically hyperbolic (*-hyperbolic) isolated G- and 1}invariant sets in a locally compact spaces has been introduced by Floer [11] in more general context. However, the notion of homological pt-hyperbolicity is enough for our considerations.
3
Hamiltonian systems
Let us first recall a general setting which will be used in our considerations. Given a Hamiltonian function Q E C 1 (JR 2m,JR) consider the Hamiltonian system of differential equations
i = J\lQ(z)
where J
=
(3.1)
[0 -I]
l O i s the standard symplectic matrix and \1 denotes the
gradient with respect to z E JR2m . We will study multiplicity results for 271'periodic solutions of (3.1) . Let us denote by H = Ht(Sl , JR2m) the Hilbert
Multiplicity results for Hamiltonian systems
157
space of 2rr-periodic, 1R2m -valued functions 00
z(t)
= ao + L (an cos(nt) + bn sin(nt»,
where ao, an, bn E 1R2m
n=1
with the inner product given by (3.2) where < a, b > denotes the standard inner product in 1R2m . Here and subsequently S1 is identified with the quotient group IRj2rrZ. Define an action of S1 on H by the time shif, Le.: (gz)(t) := z(t + g) z E H, t, 9 E S1 . Clearly, that action is linear and invariant with respect to the inner product (3.2). Hence, H is an orthogonal representation of S1 . If II "\1Q(z) II~ C1 + C2· II Z 11 8 for every z E 1R2m and some positive 5, then z(t) is a 2rr-periodic solution of (3.1) if and only if it is a critical point of the functional 4> E C 1 (H, 1R) defined by
4>(z) =
1
-2 < Lz ,z >H -¢(z),
where
< Lz,z >H=
121< < Ji(t),z(t) > dt
and
¢(z) =
121< Q(z(t»dt.
(3 .3)
(cf. [17)). It follows from the above that 4> is S1-invariant, i.e.: 4>(gz) = 4>(z) for every 9 E S1 and z E H. Hence, "\14> is an S1-map. Consequently, if z E H is a critical point of 4> then the whole orbit S1 z = {gz E H ; 9 E S1} consists of critical points. It is shown in [17] that the mapping "\1 ¢ is compact and therefore -"\14> : H --+ H is a vector field which can be written in the form -"\14>(z)
= Lz + K(z)
where K : H --+ H is completely continuous. However, K = -"\1 ¢ may not be loco Lipschitz continuous. Fortunately, it can be then replaced by a completely continuous and loco Lipschitz continuous S1-map K : H --+ H so that, J : H --+ H, J(z) = Lz + K(z) is a pseudo-gradient vector field for the functional -4> (see [6]) . Thus, with no loss we can assume, that "\14> is loc o Lipschitz continuous.
Marek Izydorek
158
Choose e1, .. . , eZ m the standard basis in IR 2m and denote Ho
= span {e1, ...
, e2m}
H:;
= span{(cos(nt))ej + (sin(nt))Jej : j = 1, ...
H;;
= span{(cos(nt))ej -
(sin(nt))Jej : j
. 2m},
n E N,
= 1, .. . ,2m},
n E N.
Clearly, H:; and H;; are S1-representations for every n E N. It is seen from (3.2), (3.3) that L is a differential operator in H which is explicitly given by 00
Lz =
L Jbn cos(nt) -
Jan sin(nt).
n=1 So, Lz
= 0 if z is a constant function Lz
={
-z
,
and
z E H:;
z , z
E
H;;
n
= 1,2, .. .
Put Hn = H:; ffi H;;, n = 1,2, ... Obviously, H = ffi~=oHn, spaces Hn are mutually orthogonal representations of S1 and Ho = ker L. For more details in non-equivariant case we refer the reader to [19] . Thus we conclude, that - V' is an S1-'cS-vector field. Moreover. ~ = (H;;+d~=o is a sequence of S1-representations which is needed to define S1-
spectra corresponding to isolating S1-neighbourhoods. Finally, if A is a symmetric 2m x 2m-matrix and Q(z) (3.1) becomes a linear Hamiltonian system
= ~ < Az, z >, then
= JAz
i
The vector field V' : H -+ H corresponding to that system preserves all spaces Hn and the restriction of V' to Hn , n ;::: 1 . may be identified with the linear map on IR4m whose matrix is Tn (A)
=[
-.!.A nJ
-J]
-~A
(3.4)
and with -A on IR2 m if n = 0 (see [19]). The following numbers have been defined by Amann and Zehnder (see [1]): 00
i-(A) := M-(-A)
+ L(M-(Tn(A)) n=l
- 2m)
Multiplicity results for Hamiltonian systems
159
00
iO(A) := MO( -A)
+
L MO(Tn(A)) n=1
where M- (U) is the number (with multiplicity) of negative eigenvalues of a symmetric matrix U and MO(U) is the dimension of its kernel. In the sequel we will use SI-equivariant Borel cohomologies with real coefficientes. We recall, that H*(BS1,IR)::::::J IR[w], w E H2(BS1,IR). Denote by ", a local flow generated by - V 4? 3.1. Proposition. Assume that Q(z)
> 0,
D(r) := {z E Hj
with Inv(D(r),,,,) PROOF:
is linear
II Vq(z) 11= o(lzl)) as z -+ O. If iO(A) = 0 then for sufficiently II z II~ r} is an isolating SI-neighbourhood for",
symmetric and small r
= ! < Az, z > +q(z), where A
= {O} and {O} is homologically pt-hyperbolic of index i-(A).
It follows from our assumptions, that an SI-CS-vector field - V4? :
H -+ H has a derivative Ao at 0 E H defined by (3.4) which is an isomorphism since iO(A) = O. Thus, the origin in H is an isolated invariant set and D(r) is an isolating SI-neighbourhood for TJ with Inv(D(r), TJ) = {O} if r is sufficiently small and positive. Moreover, the SI-CS-index h~~(D(r),TJ) is the homotopy type of a spectrum E = E(~) such that (i) for each n ~ 0, En is a one-point-compactification of a certain finite-dimensional representation of SI and therefore it is a spherej (ii) En+l = SH;:+l "En and dim En = i-(A) +2mn > 0 for sufficiently large n, say n
~
r.
Consequently, by Remark 2.6
HSI (E, IR) Since dim Er = i- (A) modules
::::::J
H;tP(r)(Er, IR) = H;t 2rm (Er , IR).
+ 2mr
there is the suspension isomorphism of IR[w]-
where So stands for zero-dimensional sphere with a base point. Hence,
which proves that {O} is homologically pt-hyperbolic of index i-(A).
Marek Izydorek
160 3.2. Proposition. Assume that Q(z)
= ~ < Az, z > +q(z)
where A is linear
symmetric and \7q(z) is bounded. If iD(A) = 0 then there exists a maximal bounded isolated invariant set T for.". It is homologically pt-hyperbolic of index i-(B).
The proof is similar to that of Proposition 3.1. The following theorem has been proved in [15] . 3.3. Theorem. Let." be a flow generated by a gradient 5 1-CS-vector field f : H ~ H, i. e.: f = \7 h for some 5 1-invariant functional h : H ~ lit Assume that:
(1) {O} cHis homologically pt-hyperbolic isolated invariant set for
'T/ of
index 2p E Zj (2) there is a homologically pt-hyperbolic isolated (G- and .,,-) invariant set T for." of index 2q E Z and 0 E Tj (3) if x E Tn f- 1(0) and x =J. 0 then its isotropy group G", =J. 51. Then, except 0 there are at least I p - q I orbits in Tn f-1(0).
As a direct consequence of the above Theorem and Propositions 3.1 and 3.2 we obtain the following. 3.4. Theorem. Let A, B be symmetric, 2m x 2m-matrices. Let Q E C1 (JR 2m , JR) satisfy the following conditions: • Q(z) = ~ \7qD(Z) 11= • Q(z) = ~ bounded;
<
Az, z
0(11 z III
> +qD(Z) in a neighbourhood of 0 E as z ~ OJ
and
II
< Bz, z > +qoo(z) in a neighbourhood of 00, and \7qoo is
• 0 E ]R2m is the unique critical point of Q.
If iD(A)
]R2m,
= iD(B) = 0, then system (3.1) possesses at least 1
2 I i-CAl nonconstant 27r-periodic solutions.
i-(B)
I
Multiplicity results for Hamiltonian systems
161
The above theorem generalizes Costa and Willem result from [7] obtained for positive definite matices A, B and for strictly convex Hamiltonian functions (see also [16]). In particular, 0 E JR2m is always a global minimum of Q. In our case it may be a saddle point. In what follows U stands for an open halfline (0,00), where 0 is negative or -00. Consider a map F : U -+ JR of class C 1 such that
(F.l) F'(x)
i
0 for each x E U ;
(F.2) there exists lim",-+oo F'(x) (finite or infinite); (F.3) there exists F"(O). Let B : JR2m -+ JR2m be a linear isomorphism. We will be concerned with Hamiltonian functions Q : JR2m -+ JR,
Q(z) Since the case lim",-+oo F'(x)
=a i
= F(II B
11 2 ) .
(3.5)
0 is rather standard we discuss the most
interesting extreme cases only, i.e. lim F'(x) = 0 and lim F'(x) = x-+oo
x~oo
±oo.
Our first theorem states as follows.
3.5. Theorem. Let a Hamiltonian function Q be of the form {3.5} with F satisfying lim",-+oo F'(x)
= O.
Then, system {3.1} possesses at least
. ! I i-(2F'(0)BT B) - 2m I if F'(O) > 0, . ! I i-(2F'(0)BT B) I if F'(O) < 0, nonconstant 27r-periodic solutions. PROOF:
By our assumptions Q is continuously differentiable in JR2m and
VQ(z)
= 2F'(11 Bz 112)BT Bz,
where BT denotes the map conjugated to B. Since for each x E U F'(x) one has
VQ(z)
= 0 iff z =
O.
i 0,
Marek lzydorek
162
Furthermore, the vector field \1Q : JR2m -+ JR2m is assymptotically linear, i.e. admits derivatives at the origin and at the infinity. The derivative at 0 E JR2m
is selfadjoint and positively or negatively definite , depending on a sign of
F'(O). We claim that the derivative at the infinity Aoo = 0 so that, we have a resonance. It is enough to show that lim z-+oo
II \1Q(z) II = o. II z II
Indeed, we have
0< lim 112F'(11 Bz 112)BT Bz II < 2 II BT B II lim IF'(II Bz 112)1 II z II
II z II
- z-+oo
-
II z II
Z-+oo
= o.
Given f3 E JR such that F'(x) = f3 for some x> O. Set y = max{x E U; F'(x) f3} and define F{3 : U -+ JR, Ii: x _ {
{3()-
F(x) F(y)+f3(x-y)
=
if x:::; y if x> y
Then, F{3 is a CI-map and if x:::; y if x> y The gradient of the modyfied Hamiltonian function Q{3(z) given by the formula
\1Q (z)
{3
={
\1Q(z) 2f3BT B
if if
= F{3(11
Bz
II Bz 112:::; Y II Bz 112> y
W)
is
(3.6)
Obviously, \1Q{3(z) is assymptotically linear and its derivatives at 0 E JR2m and at the infinity are equal to Ag respectively. Since limx--+oo F' (x) choose f3 so close to 0 that
iO(A~) Moreover, as f3F'(O)
= Ao = 2F'(O)BT B
= 0 and BT B
and A~
= 2f3BTB,
is positively definite one can
= 0 and i-(A~,) = M-(-2f3B T B).
> 0 one has M-(-2f3B T B)
= M-(-2F'(O)BTB).
Multiplicity results for Hamiltonian systems
163
If iO(Ao) = 0 then by Theorem 3.4 the Hamiltonian system i
= J\1Q/3( z )
(3.7)
possesses at least
nonconstant 2rr-periodic solutions. As iO(A~) = 0 the system
does not have nonconstant 2rr-periodic solutions and therefore each 2rr-periodic solution of (3.7) is a solution of (3.1). Finally, let us notice that M-( -2F'(0)B T B) is equal to 0 if F'(O) < 0 and is equal to 2m if F'(O) > O. There are many examples among classical functions satisfying assumptions of Theorem 3.5. As a map F(x) one can choose for instance: ce-"', cln(1 + x), carctg(x) , c(1
+ x)-S
with s > 0 or - 1 < s < 0, ect.,
where c -:F O. Note, that if F is one of the above F(x 2 ) is non-convex so that,
Q(z)
= F(II Bz 112) in non-convex as well. Choose F(x) = ~e-'" and B = aId, where Id is the identity map and k < a 2 < k + 1 for some kEN U {O} then
EXAMPLE :
in
1R2m
Q(z) = !e-lIazIl2 2
and system (3.1) possesses at least k . 2m nonconstant 2rr-periodic solutions.
3.6. Theorem. Let a Hamiltonian function Q be of the form (3.5) with F satisfying lim"'-too F'(x) = 00. Then, system (3.1) possesses infinitely many nonconstant 2rr-periodic solutions. We use similar methods as in the proof of Theorem 3.5. This time however, \1Q is not asymptotically linear. By our assumptions F'(x) > 0 for every x E U. Since the Hessian of Q at 0 E 1R2m , A o = 2F' (O)BT B is positively definite it follows from (3.4) that there is a increasing sequence of real numbers PROOF:
«(3n)'f such that: • limn-too (3n =
00 ;
Marek Izydorek
164 • iO(f3nBT B)
= 0 for every n
• lim n -+ oo i- (f3nBT B)
E N;
= 00.
By Theorem 3.4, if iO(A o) = 0 then for each n E N the Hamiltonian system (3.8)
i = J"lQ(3n (z)
possesses at least
!2 I i-(Ao) -
i-(f3n BTB ) I
nonconstant 21T-periodic solutions, where Q(3 is defined by (3.6) . The condition iO(f3nBT B) = 0 implies that all 21T-periodic solutions of (3.8) are also solutions of (3.1). Since lim n -+ oo li-(f3nBTB)1 = 00 our assertion is concluded. The case lim x -+ oo f'(x) = -00 is similar. As an example one can choose f(x) = (1 + X)8 with s > 1, f(x) = eX ect. Clearly, there are also functions for which the corresponding Hamiltonian
x3
function is non-convex, e.g.: f(x) = 2x - 3!
x5
+ ST·
References [1) H. Amann and E. Zehnder, Periodic solutions of asymptotically linear Hamiltonian systems, Manuscr. Math. 32, (1980), 149-189. [2) T . Bartsch, "Topological methods for variational problems with symmetries"; Lecture Notes in Math. 1560 Springer-Verlag, Berlin, Heidelberg, 1993. [3) T . Bartsch and M. Clapp, Critical point theory for indefinite functionals with symmetries, J. F'unct. Anal. 138 No .1, (1996), 107-136. [4] T . Bartsch and M. Willem, Periodic solutions of non-autonomous Hamiltonian systems with symmetries, J. Reine und Angew. Math. [5) V. Bend, On critical point theory for indefinite functionals in the presence of symmetries, Trans. AMS 274, (1982), 533-572.
[6) K.C . Chang, "Infinite Dimensional Morse Theory and multiple Solution Problem", Birk- hauser, Boston, (1993) . [7) D.G. Costa and M. Willem, Lusternik-Schnirelman theory and asymptotically linear Hamiltonian systems, Colloquia Math. Soc. J . Bolyai vol. 47 Differential Equations: Qualitative theory, Szeged Hungary 1984, North Holland 1986, 179191.
Multiplicity results for Hamiltonian systems
165
[8] T. tom Dieck, "Transformation groups", W . de Gruyter and Co., Berlin, 1987. [9] E. Fadell and P. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math. 45 (1978) 139-174. [10] G. Fei, Relative Morse Index and Its Application to Hamiltonian Systems in the Presence of Symmetries, J . Diff. Equat., 122, (1995), 302-315. [11] A.Floer, A refinement of the Conley index and an application to the stability of hyperbolic invariant sets; Ergod. Th. & Dynam. Sys. 7 (1987), 93-103. [12] A. Floer and E. Zehnder, The equivariant Conley index and bifurcations of periodic solutions of Hamiltonian systems, Ergod. Th. & Dynam. Sys. 8' (1988), 87-97. [13] K. G-:ba, M. Izydorek and A. Pruszko, The Conley Index in Hilbert Spaces, Studia Math. 134 (3), (1999), 217-233. [14] M.lzydorek, A cohomological Conley index in Hilbert spaces and applications to strongly infinite problems, accepted in J . Diff. Equat. [15] M.lzydorek, Equivariant Conley index in Hilbert spaces and applications to strongly indefinite problems, submitted. [16] J. Mawhin and M. Willem, "Critical point theory and Hamiltonian systems" , Berlin Heidelberg New York, Springer-Verlag, (1989) . [17] P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Reg. Conf. Ser. Math., vol 35, Providence, R.I., Am. Math. Soc. (1986). [18] E. H. Spanier, "Algebraic topology", New York, Mc Graw-Hill 1966. [19] A. Szulkin, Cohomology and Morse theory for strongly indefinite functionals, Math. Z. 209, (1992), 375-418. [20] Z.Q.Wang, Equivariant Morse theory for isolated critical orbits and its applications to nonlinear problems; Lect. Notes in Math. 1306, Springer-Verlag (1988),
202-221. Technical University of Gdansk, Faculty of Technical Physics and Applied Mathematics, 80-952 Gdansk, ul. G. Narutowicza 11/12, Poland current address: Universitat Rostock, Fachbereich Mathematik, Universitatsplatz 1, 18055 Rostock, Germany email: [email protected]
The Dynamics of the Flow for Prescribed Harmonic Mean Curvature * Huai-Yu Jian and Bin-Heng Song Department of Mathematical Sciences Tsinghua University, Beijing 100084, P.R.China (e-mail: [email protected])
Abstract In this article, we study the dynamical behavior of a parabolic equation on supplement a proof of the geometric lemma in [8] and give a new and more direct proof of the apriori estimate which is essential for the existence results of convex hypersurfaces with prescribed harmonic mean curvature in [8].
sn,
1. Introduction Let M be a smooth embedded hypersurface in R n+1 and k1 , k2 , •.. ,kn be its principal curvatures. Then H- 1 is called the harmonic mean curvature of M if 1 1 1 (1.1) , H= -+ - + ... +-. kl k2 kn The question which we are concerned with is that given a function in Rn+1, under what conditions does the equation
J defined
H-1(X) = J(X),X E M has a solution for a smooth, closed, convex and embedded hypersurface M, where X is a position vector on M. The kind of such questions was proposed by S.T.Yau in his famous problem section (1). Many authors have studied the cases of mean curvature and Gauss curvature instead. See, for instance, [2, 3) for the mean curvature and [4, 5) for the Gauss curvature, and [6, 7) for general curvature functions . ·Supported by National Natural Science F'undation of China (Grant No. 19701018)
166
Flow for prescribed harmonic mean curvature
167
Let F = 1-1, then the problem above is equivalent to looking for a smooth, closed, convex and embedded hypersurface M in Rn+l such that
H(X) = F(X) , X E M
(1.2)
where H is the inverse of harmonic mean curvature given by Eq. (1.1). We are interested only in the hypersurfaces MeA, a ring domain defined by
for some constants R2 > Rl > O. For this purpose, we need to suppose that F is a smooth positive function defined in Rn+l satisfying (a) F(X) ~ nR2 for IXI = R2 and F(X) < nR1 for IXI = R 1 , and (b) F is concave in A. In [8], the first author used a heat flow method to deform convex hypersufaces to a solution to Eq. (1.2). This leads us to consider the following parabolic equation
~~ = (H(X) - F(X)) v(X) , X E Mt , t E (0, T) } X(·,O) given,
(1.3)
where X(x, t) : sn --+ Rn+l is the parametrization of M t given by inverse Gauss map, which will be solved, and v(X) is the outer normal at X E M t , so v(X(x, t)) = x by the definition. We use Mo to denote the given initial hypersurface. The following is one of the main results in paper [8]. Theorem 1.1. Suppose that F is a smooth positive function satisfying conditions (a) and (b), and the initial hypersurface Mo C A is smooth, uniformly convex and embedded, satisfyingH(Xo) ~ F(Xo) for all Xo E Mo . Then equation (1.3) has a unique smooth solution for T = 00 which parametrizes a family of smooth,closed,uniformly convex and embedded hpersufaces, {Mt : t E [O,oo)}. Moreover, there exists a subsequence tk --+ 00 such that Mtk converges to a smooth,closed,uniformly convex and embedded hypersurface which lies in A and solves problem (1 .2). There are two key facts in the proof of Theorem 1.1. One is Lemma 2.3 in [8] . We will call it Geometric Lemma. Since its proof was omitted in [8] due to the limited pages, We are going to supplement it in the next section. The other is the apriori estimate for the solution, IX(x, t)12 , of Eq. (1.3). Also see section 2 in [8] . We will give a new and concise proof for this estimate.
Huai- Yu Jian and Bin-Heng Song
168
2. A Geometric Lemma In this section, we will supplement the proof of Lemma 2.3 in [8]. Lemma 2.1 Let X be the positive vector of a smooth,closed hypersurface M in Rn+l with outer normal veX) at X E M . Then if IXI =< X, X > ~ attains a maximun R at a point Xo E M, then Xo = Rv(Xo) and
II(w, w)
~
1
/ig(w, w), Vw E TXoMj
if IXI attains a minimum r at a point Xo E M, then Xo
II(w,w)
~
= rv(Xo) and
1
-g(w,w), Vw E TXoM, r
where 9 is the metric on M and II is the second fundamental form of M with respect to the direction -v. Proof. We consider only the first case, because the latter is completely anagolous. For Xo E M and any w E TXoM, choose a curve ')'(s) on M,,,( : [0,1] -+ M, such that "(0) = X o, l' = w. Let p(X)
= IXI. Since p2(X) attains a maximum at Xo \lp2 = 0 and \l2p2
~
E M, then at this point
O.
Therefore, we have
and
~
ds 2P2 (')'(O)) = 1'(0)· \l2p2(XO)' 1'(0) + \lp2(Xo)' ;yeO) ~ O. On the other hand,
and
1 d2p2
.
2
..
2 ds2 = I')'(s) I + ')'(s) . ')'(s). Thus,
o = 2')'(0) . 1'(0) = 2Xo . w
(2.1)
and
o ~ 11'(0)12 + Xo
. ;yeO).
(2.2)
Flow for prescribed harmonic mean curvature
169
Since w E TXoM can be arbitrary, Eq. (2.1) implies
Xo
= IXolv(Xo) = Rv(Xo),
therefore
Xo . 1'(0)
=
R < v(Xo),1'(O) >
=
R < v(Xo), Dt(o)1'(O) > -R II ("y(0), 1'(0)) -R II (w, w),
which, together with Eq. (2.2), gives us that
II(w,w)
>
~h(OW R 1 R
< 1'(0),1'(0) >
1 R
< Dt(o)X, Dt(o)X >
1 "Rg("y(O),1'(O)) 1 "Rg(w,w).
3. A New Proof of the Apriori Estimate In this section, we will provide a new and concise proof of the apriori estimate for Eq.(1.3), which is essentially important for the proof of Theorem 1.1. We recall some facts in (9). Let el, e2, ... , en be a smooth local orthonormal frame on sn, and let 'Vi = 'Vei) i=l, 2, ... , n and 'V = ('VI, 'V 2,· .. , 'V n ) be the covariant derivatives and the gradient operator on sn, respctively. Since X(x, t) is the inverse Gauss map, the support function of M t is given by
>, x
E sn
(3.1)
where < ., . > denotes the usual inner product in tal form of M t is
Rn+l.
The second fundamen-
U(x, t) =< x, X(x, t)
hij(X (x,
t)) =< 'V iX, 'V jX >= 'Vi 'V jU(X, t) +c5ij u(x, t), i, j = 1,2, ... , n. (3.2)
IT M t is strictly convex, then h ij is invertible, and hence the inverse harmonic mean curvature is the sum of all the eigenvalues of the matrix bij = [ g''k hkj )-1
,
Huai- Yu Jian and Bin-Heng Song
170
where gij is the metric of M t . But the Gauss-Weingarten relation
'ViX = hikl/'V/x and the fact < 'Vix, 'Vjx
>= Oij imply gij = hikhjk.
1 H = -
1
+ ... + -
k1
kn
Therefore, bij
= ~u + nu,
= hij and (3.3)
where ~ = 2::7=1 'Vi'V i · We will often use the fact that x, 'V 1 x, 'Vzx ,··· , and 'V nX form a standard orthonormal basis at point X (x, t). It implies
X(x , t)
= u(x, t)x + 'Viu(x, t)'Vix
(3.4)
and
IX(x, tW
= UZ(x, t) + l'Vu(x, tW o
(3.5)
From now on, we assume that the initial hypersurface Mo is smooth, closed, strictly convex and satisfies H(Xo) 2: F(Xo) for all Xo E Mo. That is Uo E coo(sn) and for some positive constant Co, one has (3.6)
and ~uo(x)
+ nuo(x) - F(uo + 'ViUO'ViX) 2: 0, "Ix
E
sn,
(3.7)
Where I denotes the n x n unit matrix and Uo is the support function of Mo . The following lemma is well known (see [8]). Lemma 3.1. If for t E [0, T) with T ~ 00, X(x, t) is a solution of (1.3) which parametrizes a smooth, closed, strictly convex and embedded hypersurface, then the support functions u(x, t) of M t satisfy
~~ = ~u + nu - F(ux u(·,O) = uo(-)
+ 'ViU'ViX) , (x, t) E sn
x (0, T) }
(3.8)
and
'VZu + uI > 0 in
sn
x (0, T) .
(3.9)
Conversely, if u is a smooth solution to (3.8) and satisfies (3.9), then the bypersurface M t , determined by its support function u(x, t), is a smootb, closed, strictly convex, embedded hypersurface and solves (1.3) for t E [0, T).
Next, we sup})ose that u is the solution to Eq. (3.8) on time interval (0, T) with the initial data satisfying Eqs. (3.6), (3.7) and
Mo C A and u(x.O) = uo(x) > Rl "Ix E
sn
(3.10)
171
Flow for prescribed harmonic mean curvature We will estimate u and its derivatives. Lemma 3.2. For all (x, t) E sn x (0, T), we have
°
au(x, t) at ~.
(3.11)
Proof. Let
= ~u(x, t) + nu(x, t) -
G(x, t)
F(u(x, t)x
+ V'iUV'iX).
(3 .12)
Using Eqs.(3.4) and (3.8), we compute that for t E (0, T), aG at
= =
au
au
au ·
au
·
+ n at - Fj(X)( at Xl + V'i at V'i Xl ) ~G + nG - Fj(X)(Gxj + V'iGV'ixj), ~ at
(3.13)
°
a%f;).
where Fj = But condition (3.7) says that G(x, 0) ~ for all x E sn. This means that G(x, t) is a supersolution to the equation of type (3.13) with zero initial data. By comparison principle [10], we have G(x, t) ~ for all (x, t) E sn x (0, T). Using equation (3.8) again, we have obtained (3.11) .
°
Theorem 3.3. Under assumptions (3.6), (3.7) and (3.10), we have R~ ~ u 2(x , t)
+ lV'u(x, t)12 < R~, 'V(x, t)
E
sn
x (0, T).
(3.14)
Proof. With the aid of (3.11) and (3.10), we see that u(x, t) ~ Rl
> 0, 'V(x, t)
E
sn x [0, T).
(3.15)
This immediately implies the left hand side inequality of (3.14). Next, we will prove its right side one, i.e., u 2(x, t)
+ lV'u(x, tW < R~, 'V(x, t)
E
sn
x (0, T).
(3 .16)
For this purpose, we compute, using (3.4) and (3.8), that
+ 2nu 2 - 2F(X)u - 21V'u1 2 + 2nu 2 - 2F(X)u,
2~uu
= and
alV'ul at
2
=
~U2
(3.17)
Huai- Yu Jian and Bin-Heng Song
172
Observing that AI\7uI 2
= =
2\7i(\7i\7kU · \7kU) 21\7\7uI 2 + 2\7iUA \7i u
and using the standard formula for interchanging the order of covariant differ(see [ll]) entiation with respect to the orthonormal frame on
sn
\7 i Au
= A\7 i u -
(n - l)\7iu,
(3.19)
we obtain from (3.18) that
81~:12 = AI\7uI 2 _ 21\7\7uI 2 + 21\7u1 2 -
2\7 i uFj(X)\7 i xj .
(3 .20)
On the other hand, by (3.4), (3.2) and the fact that x, \71X,·· ·, \7 n x form a standard basis, we have < \7 i X, x >= 0 and
= =
\7 i X
< \7 i X , \7jX > \7jX h ij \7 j x
Thus, we have 2\7i UFj(X)\7i Xj
= =
Fj(X)(2\7 i u\7;\7 kU + 2u\7 kU)\7 kxj Fj(X)\7k(l\7uI 2 +U2)\7kxj.
(3.21)
Putting (3.21) in (3.20) yields 81\7u1 2
---at =
2 AI\7uI - 21\7\7uI 2 + 21\7u1 2 - Fj(X)\7 k (U 2 + l\7uI 2)\7 kxj,
which, combined with (3.17), implies
where we have used (3.5) . Now , let
f(t)
= xESn max IX(x, t)12,
t
E
[0, T) .
Were inequality (3.16) not true, then by the assumption Mo C A (see (3.10)) we could find t2 E [0, T) such that f(h) = ~ . Let
tl = inf{t E (O,T) : f(t) =
RD
Flow for prescribed harmonic mean curvature
173
and choose Xl E sn such that J(tt}
= IX(XI' tl}1 2 = R~ .
(3.23)
Obviouly, IX(XI,t}1 2 < IX(XI,tIW for all t E [O,tt}, so that we have 81XI 2 at(XI,tt} ~ O.
(3.24)
On the other hand, since (Xl, tt) is the maximum point for the function IX (x, tl W with respect to (x E sn), we have 61XI ~ 0 and V'IXI 2 2
=0
at (Xl, tt)'
Therefore, Eq. (3.22) yields 2
81XI a t
~
2u(nu - F(X}} at (XI,tt).
(3.25)
Furthermore, it follows from (3.23) and condition (a) that
F(X(XI' tt}} > nlX(xI' tt}l ~ nlu(xl, tt}l .
(3.26)
With the aid of (3.15) and (3.25), we see that (3.26) turns to 81XI 2 at(XI,tl } < 0,
which yields a contradiction with (3.24) . In this way, we have proved inequality (3.16). Acknowledgement. The first author would like to thank professors R . Bartnik, K. Tso, W. Y . Ding, and Y.D . Wang for many helpful conservations and professor B . Chow for his interests in this work.
References 1. S. T . Yau, in Seminar on differential geometry, ed. S. T . Yau ( Ann of Math Stud 102, Princeton, 1982), pp. 669-706. 2. A. Treibergs and S. W. Wei, J. Differ. Geom. 18 (1983), 513-521. 3. K. Tso, Ann. Scuola Norm. Sup. Pisa 16 (1989), 225-243. 4. V. 1. Oliver, Trans. Amer. Math. Soc. 295 (1986), 291-303. 5. K. Tso, J. Differ. Geom. 32 (1991), 389-410.
174
Huai- Yu Jian and Bin-Heng Song
6. 1. Caffarelli, J . Nirenberg and J. Spruck, in Current topics in partial differential equations, eds. K. Ohya, K. Kasahara and N. Shimakura
7. 8. 9. 10. 11.
(Kinokunize Co., Tokyo, 1986), pp. 1-26. C. Gerhardt, J. Differ. Geom. 43 (1996), 612-641. H. Jian, Science in China Ser. A 42 (1999), 1059-1066. J. I E. Urbas, J. Differ. Geom. 33 (1991),91-125. R. S. Hamilton, J. Differ. Geom . 24 (1986), 153-179. B. Chow, 1.P. Liou and D.H. Tsai, Comm. Anal. Geom. 3 (1996), 75-94.
A Symplectic Transformation and its Applications * Mei-Yue Jiang Department of Mathematics Peking University, Beijing, 100871, China R n is presented. Some applications are given, which include periodic solutions of Hamiltonian system and construction of symplectic embeddings from (R 2n , wo) to some symplectic manifolds.
Abstract A simple and elementary symplectic transformation on Tn
X
1. A Symplectic Transformation Let Tn X R n be the cotangent bundle of the torus Tn and >. = E~ Yidxi be the Loiuville form, where x = (Xl,· · ·, Xn) E Tn, each Xi is I-periodic, and Y = (YI, .. . , Yn) ERn. We consider the following symplectic form on Tn X R n,
w
= d>' + n = Wo + n,
where Wo = E~ dYi 1\ dXi is the standard symplectic form on Tn X R n, and n is a closed 2-form on Tn. Such w is called twisted symplectic form. The physical meaning of n is a magnetic filed on the torus. The Hamiltonian systems with this twisted symplectic form on the cotangent bundle was studied in the early eighties by Novikov. Some important classical Hamiltonian systems arising from physics can be written in this form, see [20]. In this article, we survey some results obtained by the author related to Tn X Rn with the twisted symplectic form w, which include the periodic solutions on a given energy surface, construction of symplectic embbedings from (R 2n , wo) to some symplectic manifolds and periodic solutions of a class of superquadratic Hamiltonian systems. All these results are based on following transformation, a simple linear algebra fact . PROPOSITION 1.1. Let n = aijdxi 1\ dXj be a constant and non-zero
E;:;
·supported by NNSF of China and Ministry of Education of China
175
Mei- Yue Jiang
176
2-form on Tn andw = wo+n. Then there exists a diffeomorphism ¢ ofTnxRn of the following form
¢ : (X, Y) ---t (x, y) , x
= X + B . Y,
y
=C .Y
where Band Care n x n matrices and det( C) -:j; 0 such that r
¢*(w)
n
= LdYit\dYi+r+ L>ijdXit\dXj +
L
dijdYit\dXi ,
(1.1)
i~2r+1
I
where 2r = rank(A), and (d ij ) is a constant matrix. Proof: The matrix A = (aij) is non-zero and anti-symmetric, so the rank of A is an even number 2r . Let be the r x r identity matrix and
Ir
J =
(0 -Ir) . Ir
,
0
For a matrix B = (b ij ), we write Bdx t\ dy tions show that
¢*(w) = AdX t\ dX
= L:ij bijdxi t\ dXj.
+ (2A · B + C)dY t\ dX + (Bt
. A- B
Simple calcula-
+ Bt . C)dY t\ dY,
where the upper t denotes the transpose of a matrix. The matrices B and C can be obtained as follows . Let B be an invertible n x n matrix such that
Bt . A . B
= ( ~,J2r' ~)
and the matrix C be determined by
Bt . C = ( - ~J2Tl 0,
0 ) . I n - 2r
Then it is easy to see that Band C satisfy (1.1) . As a corollary, we have following result, see also [6]. COROLLARY 1.2. Let n = L:~.j aijdxi t\ dXj be a constant symplectic form on T2n and w = Wo + n. Then there exists a diffeomorphism ¢ of T 2n X R 2n of the following form
¢ : (X, Y) ---t (x, y), x
= X + B . Y,
y
=C .Y
where Band Care 2n x 2n matrices and det( C) -:j; 0 such that 2n n ¢*(w) = L dYi t\ dYi+n + L aijdXi t\ dXj = WI . I
(1.2)
I
On the right hands of (1.1) and (1.2), the first term L:~ dYi t\ dYi+r and L:~ dYi t\ dYi+n are indepedent of other variables, this point is crucial for the applications later.
A symplectic transformation and its applications
177
2. Symplectic Capacity and Periodic Solutions In this section, we study the existence of periodic solutions of Hamiltonian systems on a given energy hypersurface. In order to state our results, we need a symplectic capacity which is defined by Hofer and Zehnder as follows. Let (N,w) be a symplectic manifold, we denote ll(N,w) the set offunctions H on N satisfying (1) there is an open nonempty set U C N such that H(x) = 0 for x E U; (2) there are compact set K C int(N) and positive number m(H) such that H(x) = m(H) for x E N \ K; (3) 0 ~ H(x) ~ M(H) for x E N. For a function H on N, let XH(x) be the Hamiltonian vector field defined by w(XH ,·) = dH( ·). DEFINITION 2.1. HE ll(N,w) is called admissible if :i;
= XH(x)
has no nonconstant T-periodic solution for 0
= sup{m(H)IH E ll(N,w),
H
c(N,w)
~
1. Let is
admissible} .
This number c (N, w) E [0, +00] is called symplectic capacity. It is a symplectic invariant. For main properties of this invariant, we refer to [9]. Based on proposition 1.1, we can prove the following theorem. THEOREM 2.2. Let 0 = L aij(x)dxi 1\ dXj be a closed non exact 2-form and w = d>" + O. Then for any open subset U of T*Tn with compact closure, we have c(U, w) < +00. This theorem has some consequences for periodic solutions of autonomous Hamiltonian systems (HS) i = XH(z), Z = (x, y) on a given hypersurface E = {(x,y)IH(x,y) = c}. E is called regular if
E
= {(x, y) E Tn
X
Rn, H(x, y)
= c}, dH(x, y) :f 0
for
(x, y) E E.
Note that the existence of periodic solution of (HS) on E depends only on the hypersurface E. THEOREM 2.3. Let 0 be as above and E be a compact hypersurface of contact type of (Tn X R n, d>" + 0) . Then there exists a periodic solution of (HS) on E which is contractible in Tn X Rn provided E is regular. Let H be a smooth Hamiltonian function on Tn xRn such that for all h,Eh = ((x,y) E Tn X Rn,H(x,y) = h} is compact. Set
C
= {h E R,dH(x,y):f 0
for
(x,y),H(x,y)
= h} .
178
Mei- Yue Jiang
Then for almost all h E C, Hamiltonian equation (HS) has a periodic solution on 'Eh which is contractible in Tn X R n. A compact hypersurface 'E is of contact type, if there is a vector field ~ defined in a neighborhood of 'E which is transversal to 'E and satisfies L{w = w. Remark 2.4. Theorem 2.2 has been known to experts for some time. IT the cohomology class of n is zero, it is proved in [10]. In this case, the periodic solution obtained may not be contractible in Tn X Rn. IT the cohomology class of w is rational, it can be deduced from a result in [11]. The case that n is nondegenerate, hence n must be even, is proved in [6]. The formulations as stated is proved in [13], see also [8]. For a generalization, see [19]. Remark 2.5. There are some results for n = 2, see [7], [21] and [25]. In particular, following result is proved by Arnold and Kozlov. Let n = a(xI, X2)dxIl\dx2, a(xI, X2) =I 0 for any (Xl, X2) E T2. Then for h > > 1, there are three periodic solutions of (H S) satisfying H(x, y) = 2:; IYil 2 + V(x) = h. The proof relies on the well known symplectic fixed point theorem due to Conley-Zehnder, see [15]. Proof of Theorem 2.2: Since n a closed, non exact 2-form on Tn, by the de Rham theorem, there are constants Cij, 1 ~ i,j ~ n and I-form 0: on Tn such that n = Cijdxi 1\ dXj + do:
t
L i,j
iTn aij(x)dx,Cij =I 0 for some i,j, since {dXi I\dxjh
with Cij = H2(Tn). Let we have
0:
= 2:~ ai(x)dxi
with each ai(x) being a function on Tn, then
w = 'EfdYi 1\ dXi + n = 'Efd(Yi + ai(x» 1\ dXi
Set
n
W=
L dYi I
1\
dX i +
+ 'Ei,jCijdxi 1\ dXj.
L cijdXi
1\
dXj ,
i,j
(2.1) Then the map (x,y) -t (x,y+a(x» = (X, Y) with a(x) = (al(X),'" ,an(x» is a symplectomorphism between (Tn X Rn,w) and (Tn X Rn,w), which maps a bounded open set of (Tn X R n, w) to that of (Tn X R n, w). Thus in order to prove theorem 2.2, it is enough to prove that c(U,w) < +00 for any bounded open set U. By proposition 1.1, we only need to show for such U, C(U,WI) < +00 where r
WI
= L dYi 1\ dYi+r + L cijdXi 1\ dXj + 1
L
i>2r+1
dijdYi
1\
dX j .
A symplectic transformation and its applications
179
This follows from PROPOSITION 2.6. Let (N,W2) be a compact symplectic manifold without boundary such that the second homotopy group 11"2 (N) = {O} and let B 2r(R) be the ball of radius R in R 2 r = (Y1 ,· • . , Y2r ). Set M = B2r(R) X N, = E~r dYi I\dYi+r +W2 . Then c(M,no) = 1I"R2.
no
For a proof of this proposition, see [9]. By assumption, U is a bounded open set of
hence it is contained in the set (Tn X Bn-2r(R) X B 2r(R),wd for some R. Now we take an integer k > 2R, let Tn-2r(k) = Rn-2r /(kz n - 2r ), where zn-2r is the integer lattice in R n-2r . Then we can find a symplectic embedding from (Tn X Bn-2r(R) X B 2r(R),Wl) into (Tn X Tn-2r(k) X B2r(R),W2 + E~r dYi 1\ dYi+r), where n
W2
= 2: cijdXi 1\ dX j +
is a symplectic form on Tn concludes that
2:
dijdYi /\ dXj
i~2r+l
1 X
Tn-2r(k). Thus the monotonicity of the c(N,w)
C(U,Wl) ::::; 1I"R2
< +00.
This finishes the proof of theorem 2.2. The existence of a periodic solution of (H S) on ~ which is a compact regular hypersurface of contact type and the almost existence result follow immediately from finiteness of c(U, w), see [9]. The contractibility of the periodic solutions follows from the fact that we in fact consider (HS) as a Hamiltonian system on R 2n , the periodic solutions obtained are periodic on R 2n, therefore they are contractible on Tn X R n.
3. Construction of Symplectic Embeddings Let (R2n, WO), Wo = E~ dYi /\ dYHn be the standard symplectic space. We consider in this section the existence of symplectic embeddings from (R2n , wo) or (B 2n(r),wo) into other symplectic manifolds. It is related to the following non-squeezing theorem of Gromov, see [17]. THEOREM. Let B2n+2(r) be the ball of radius r in R 2n+2 and (M2n,w) be a 2n-dimensional symplectic manifold. If there is a symplectic embedding
then r ::::; R.
Mei-Yue Ji
180
However, it is shown in [16] that the disk (B2(R),wo) can not be repla by the 2-dimensional torus (T2, w). In fact, Polterovich showed that there symplectic embedding from (R4,wo) to (T2 x R 2,w EB wo), see [18]. Our here is to show the existence of symplectic embeddings from (R2n , wo) i some symplectic manifolds by elmentary methods. The following theorems proved in [12] . THEOREM 3.1. Let ~ be a closed surface of genus g 2: 1 and let w I symplectic form on ~. Then there is a symplectic embedding if> from (R 4 , to (~x R2,wEBwo). THEOREM 3.2. Let w be a constant symplectic form on T2n. Then the7 a symplectic embedding if> from (R 2n+2m,wo) into (T2n X R2m,wEBwo) . The proof of theorems is quite elementary and the symplectic embeddi can be constructed explicitly. It is based on the proposition 1.1 and the that the universal covering of ~ is R 2 and that of T 2n is R 2n. For any two ( forms WI and W2 on ~, by Moser's theorem, there are diffeomorphism if> ( and constant c such that if>*(wd = c· W2, see [9]. Thus we can fix a symple form w as follows. If ~ = T2, then we set w = dXl 1\ dX2, if the genus ( 9 2: 2, then set w be the area form of the metric of curvature -1. In follov we use Xi, Xi to denote the coordinates of ~ or T 2 n, and Yi, Yi denote tha R2n. Let U(a, b) = (a, b) x R, X E (a, b), y E R, Wo = dy 1\ dx . The follov simple geometric fact will be used. LEMMA 3.3. Let U be a contractible domain ofR2 such that I fuwol =-1 then (U, wo) is symplectomorphic to (R 2 , wo) . Proof of theorem 3.1: By proposition 1.1, there is a symplectomorpsim
if> : (T2
X
R2, wEB wo) -+ (T2 x R2, dYl
1\
dXl
+ dY2 1\ dX2 + dXl
1\
dX2)'
xR2 ),
On the open subset U(O, 1) x U(O, 1) of (T2, the twisted symplectic £ is symplectomorphic to Wo EBwo . Thus the restriction of if>-l to U(O, 1) x U(f produces a symplectic embedding from (U(O, I) x U(O, 1), Wo EB wo) to (1 R2, wEB wo). This proves theorem 3.1 for the case ~ = T2 by lemma 3.3. Now we assume the genus of ~ g 2: 2. Let w be the area form of the mE on ~ with curvature -1. The universal covering of ~ is the hyperbolic Sl (H2, ds 2 ), where
H2
= {(X 1, X)2
E R2 ,X 2
> O} ,s d 2 -_ dXf X2 + dX? 2
Let P be the covering map, then P*(w) = dX';!X For 8 > 0, cons 2 if>l: U(-8,8) x U(-8,8) -+ H2 x R2, (Xl,X2,Yl,Y2) -+ (X l ,X2,Yi,Y2), 2 •
Xl = -e(Yl-:l:2) (Xl Y 1 = Yl,
+ Y2), X 2 =
e(Yl-:l:2),
Y2 = -Y2,
A symplectic transformation and its applications
181
then
4>i(P*(w) ffi wo) = W2 = dXI 1\ dX2
+ dYI 1\ dXI + dY2 1\ dX2·
For <5 > 0, set
Uo = {(x,y) E R2, Ixi < 6, IxYI < 6}. Let 4>2 : U ( -6, <5) x Uo -+ 1: x R 2 be defined by
4>2(XI,X2,YI,Y2) = (P(X I (X,y),X 2(x,Y)),YI,-Y2). It can be shown that for small 6 > 0, 4>2 is a symplectic embedding from (U(-6,6) x Uo,W2) to (1: x R2,W ffiwo) . Now we fix a such 6 and show that (R\wo) and (U(-<5,6) x Uo,WI) are symplectomorphic, this will complete the proof of theorem 3.1. Noting
is a symplectomorphism from (U(-6,6) x Uo,W2) to (U(-6,6) x Uo,wo ffiwo), which is symplectomorphic to (R\wo) by lemma 3.3. This finishes the proof of theorem 3.1. Next we give a sketch of proof of theorem 3.2. The part n :::; m is an easy consequence of proposition 1.1. We assume n = m. n < m follows from this easily. By proposition 1.1, it suffices to show that there is a symplectic embedding 4>3 : (R4n,wo) -+ (T2n X R2n,wd. Consider the set
{(X,Y) E T 2n
X
R2n,0
< Xi < 1,li = 1, ··· ,2n},
on this set we have WI = L~n d(Y + AX) 1\ dX and (X, Y) -+ (X, Y + AX) is a diffeomorphism, thus it is symplectomorphic to n products of U(O, 1) with symplectic form Wo = L~n dY; 1\ dX i , which is symplectomorphic to (R4n,wo). This proves theorem 3.2 if n = m. The remaining part of theorem 3.2 can be proved as follows . We assume m = 1, n = 2, the other case is similar. It is enough to show that for small 6, there is a symplectic embedding from symplectic manifold (U(0,6) x U(0,6) x U(O, 6), Wo ffi Wo ffi wo) to (T4 x R2, W ffi wo) . This will be completed by two steps. Let P : R4 -+ T4 be the covering map, we first find a symplectic embedding IJ! from this manifold to (R4 x U(0,6),wo ffiwo). Then choosing a linear symplectomorphism S : (R 6 , wo) -+ (R4 x R 2 , W ffi wo) properly, we can show that (P, I dR 2 )oSolJ! is a needed symplectic embedding to (T4 XR2, wffiwo). The map IJ! can be given explicitly by following way. Take a diffeomorphism ! : R -+ (0,6), set 4>4: (U(0,6) x U(0,6),wo ffiwo) -+ (R2 X U(0,6),wo ffiwo),
4>4(Xl,Yl,X2,Y2)
= (-Y2 -Xl,Yl,!(Y2),
+ X2) !'(Y2) ).
YI
182
Mei- Yue Jiang
Let 111 = (Id U (O ,o),¢4) 0 (¢4,Id u (o,o)). The linear transformation S is given by following lemma. Let {el ,· ·· ,e2n} be the standard orthogonal basis of R2n . Recall that a symplectic basis for w = L bijdxi 1\ dXj is a basis {II,· .. , hn} such that w(j2i-I,hi) = 1 = -w(j2i,hi-d , i = l,···,n,
w(fi,!i)=O for other i,j. LEMMA 3.4. Let w = L bijdxi 1\ dXj be a constant symplectic form on R2n. Then after a permutation of the basis {eI,· · ·,e2n}, still denote it by {el, . .. , e2n}, there is a symplectic basis {II, ·· ·, hn} for w such that Ii E span{el,··· ,ed· This lemma can be proved by induction on n, see [13] . Let (II, · .. , hn) = T· (eI, . . . ,e2n) and let S be the linear transformation defined by
Then S·w = Wo = L~ dY2i-1 1\ dY2i and S considered as made up of 2 x 2 blocks, is a block of upper triangular matrix. For w EEl Wo on R 6 , we have a such matrix S = (Sij) . Then elementary computations show that for 8 small, (P,Id R 2) 0 S 0 111 U(O,8) x U(O,8) x U(O,8) -+ T4 x R2 is an embedding satisfying This completes the proof of theorem 3.2. As a consequence of above construction, we can obtain some estimates of another symplectic capacity, the Gromov's capacity, which is defined by cI(M,w)
= SUp{11T2 , 3a
symplectic embedding
¢ : (B 2n (r) , wo) -+ (M,w)} .
THEOREM 3.5. Let w be a constant symplectic form on T2n , and (~, r) be a closed 2-dimensional symplectic manifold of genus g ~ 1. Then there is a constant C > 0 such that for R > > 1, the following inequalities hold: (1) cI(T 2n X B 2(R) , w EEl wo) ~ C · Rn~l . (2) CI (~ x B2(R) , rEEl wo) ~ C· log(R) if the genus of ~ g ~ 2. REMARK 3.6. It is easy to see from comparison of the volume that the exponent n!l in (1) is optimal. For the standard 2n-torus (T2n , wo), Wo =
L~ dX2 i-1 1\ dX2i, we can take C = (~)n~l . For any 2-dimensional symplectic manifold (~, r) with 1IE rl = 1r R 2, since for any r < R, there is a symplectic embedding from (B2(r), wo) to (~, r), so
A symplectic transformation and its applications
183
monotonicity of cl(M,w) implies
COROLLARY 3.7. Let (E, r) be 2-dimensional symplectic manifold, then if
4. Periodic Solutions of Nonautonomous Hamiltonian Systems In this section, we consider periodic solutions of Hamiltonian systems on Tn X Rn for time dependent Hamiltonian H which is periodic in the variable t with period 1 with the twisted symplectic form. We assume that n = " .. aiJ·dxi 1\ L.-i'l. ,) dx j is a constant symplectic form . Thus the Hamiltonian system can be written as (4.1) Ax - iJ = Hz(t , x, y). x = Hy(t, x, y), It will be convenient to consider (4.1) as a Hamiltonian system on R 2n with the Hamiltonian H being periodic in variables Xi, i = 1,2" . . ,n with period 1 later. Since Conley-Zehnder's solution of the Arnold conjecture on torus, see [4], there have been many works on existence of periodic solutions for spatially periodic Hamiltonian syetems, either on torus or on Tn X R n with A = 0, we refer to [2], [3] and the references theirin. Our aim here is to prove following THEOREM 4.1. Let H E C1(R X T 2n X R2n) satisfy (H1) H(t, X, y) = H(t + 1, X, y), H(t, X, y) = H(t, -X, -y); (H2) there are constants IL > 2, Ro > 0 such that
Hy(t,x , y)· Y 2: ILH(t,x,y) > 0
for
(t,x) E R
X
T 2n , Iyl 2: Ro;
(H3) IHzl ~ M for some constant M > 0,(H4) H 2: 0 and H(t, X, y) = o(lxl 2 + IYI2) near (0,0),(H5) there are constants Cl, C2 > 0 such that IdH(t, X, y)1
~ Cl
+ c2Hy . Y
for
any
(t, X, y) .
Then there is a sequence of 1-periodic solutions (Xj(t), yj(t)) of (4 .1) which are contractible in T 2n X R 2n . There are many results as above for variational problem, i.e. evenness and a kind of superquadraticity ensure infinitely many solutions. The case A = 0 has been considered in [2]. For the case A = 0, one can expect existence of noncontractible periodic solutions in other homotopy classes, however, simple examples show that this is impossible if A i- O. The reason for this phenomenon
Mei- ¥ue Jiang
184
is that (4.1) is equivalent to a Hamiltonian system on T 2n X R 2n with a spliting symplectic form by proposition 1.1. This enables us to use some existences results for periodic solutions of Hamiltonian system on R 2n with slight modifications . PROOF OF THEOREM 4.1: By proposition 1.1, (4.1) is equivalent to A~
= K~(t,~, 1]),
J17
= K.,,(t,~, 1])
(4.2)
with K(t,~, 1]) = H(t, ~ + B1], C1]), and J is the standard symplectic matrix on R 2n. It is easy to see that the new Hamiltonian K satisfies same conditions as in theorem 4.1 if (x,y) is replaced by (~,1]). We will show that there is a sequence of I-periodic solutions (~i,1]i) of (4.2) such that
where I(~,1]) =
(II.
10
(2(A~,~)
1
+ 2(J17,1]) -
K(t,~,1]))dt.
(4.3)
For simplicity, we assume A = J, the standard symplectic matrix on R 2n and the Hamiltonian H satisfies (H6) IdH(t,x,y)1 ::; C3 + c41ylS for some constants C3,C4 and s. This condition can be removed by a trunction procedure as in [23). With this condition, the functional I in (4.3) is C 1 and even on the space (~, 1]) E X = H t (Sl , R 2n) X H t (Sl , R 2n). The proof given below is a slight modification of that in [23], therefore it will be sketchy. Let X = X_ EB Xo EB X+, where X_, X+ and Xo are the negative, positive and zero space of the quadratic form
We know Xo = R 2n X R2n . (HI) implies that I is well defined on X_ EB (T2n X R2n) EB X+. LEMMA 4.2. The functional I(~, 1]) satisfies the (P.S) condition on X_ (B (T2n X R2n) EB X+ . Proof: Let (~I' 1]1) be a (PS) sequence, it is easy to show that lI~dl is bounded modulo the translation of the integers. With this fact in hand, the boundedness of II1]dl and a convergent subsequence can be proved as in [1). Now we consider I as a functional on X, (P.S) condition fails, however, the deformation lemma holds for I on X because of above lemma, thus critical point theorems can be applied.
A symplectic transformation and its applications
Let S be the family of sets A w.r.t. O. For A E S, let
c
185
X \ {O} such that A is closed and symmetric
')'(A) = min{jI3¢ E C(A,Rj \ {O}),¢(-x) = -¢(x),x E A}, this is the Z2 index. For properties of this index, we refer to [24]. Let
By (H2), there are constants
C5, C6
such that
hence we get (4.4) Since dimVk n X+ rk > 0 such that
= 2kn and J.L >
2, (4.4) and (H4) imply that there is an
I(~,"1) ::; 0 for z E Vk , IIzil ~ rk . 2 As H(t, x, y) = o(lxl + IYI2) near (0,0), we have K(t,~, "1) (0,0), therefore for small p > 0,
(4.5)
= o(I~12 + 1"112) near (4.6)
for some constant a . Let B(r) be the closed ball of radius r in X centered at O. Set Dk = B(rk) n Vk . Let P_ be the projection from X to X_ and let G k be the class of maps h E C (D k, X) with following properties;
(1) h( -x) = -h(x); (2) h(z) = z for z E 8D k ; (3) P_h(z) = a(z)L + g(z), where 9 is compact and a E C(Dk' [1, aD, a depending on h. Finally for j EN, set
rj
= {h(Dk \ Y)lk ~ j, hE G k , YES, Cj
and ')'(Y)::; 2nk - 2nj},
= infBEr;suPzEBI(z).
The conclusion that {Cj} is an unbounded sequence of critical values of I follows from the standard deformation argument. This completes the proof of theorem 4.1.
186
Mei- Yue Jiang
Remark 4.3. In order to use the Z2 mountain pass theorem to obtain an unbounded sequence of critical values for an even functional, one usually need to verify that (4.6) holds and (4.5) holds for X_ EEl Xo EEl Ek, where Ek is an arbitrary sequence of finite dimension subspaces of X+ such that dim(Ek ) -t +00. However, the same conclusion hold if for one such sequence, (4.5) holds, see [5] (corollary 7.23) if dim(X_ EElXo) < 00. In our case, dim(X_ EElXo) = 00, this can be treated by a Galerkin approximation. Remark 4.4. If we assume the growth condition (H6), then (H5) can be dropped. (H5) is only needed in the trunction. If we consider autonomous Hamiltonian system, that is H is independent of time t, then we can use SI symmetry instead of Z2 symmetry. In this case, conditions (H5) and symmetric condition are not needed.
References [1] A. Bahri and H. Berestycki, Forced vibrations for superquadratic Hamiltonian systems, Acta Math., 152 (1984), 143-197. [2] T. Bartsch and Z. Q. Wang, Periodic solutions of spatially periodic, even Hamiltonian systems, J . Diff. Equa., 135 (1997), 103-128. [3] K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhiiuser, Boston, 1993. [4] C. C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture by V. I. Arnold, Invent. Math., 73 (1983), 33-49. [5] N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory, Cambridge University Press, 1993. [6J V. 1. Ginzburg, On closed trajectories of a charge in a magnetic field. An application of symplectic geometry, in Contact and Symplectic Geometry, ( C. B. Thomas ed.), pp131-148, Cambridge University Press, 1996. [7] V. L. Ginzburg, On the existence and non-existence of closed trajectories for some Hamiltonian flows . Math. Z. (1996), 223: 397-409. [8] V. L. Ginzburg and E. Kerman, Periodic orbits in magnetic fields in dimension great than two, preprint, 1999. [9J Hofer H. and Zehnder E. Symplectic Invariants and Hamiltonian Dynamics. Boston Basel, Birkhiiuser, 1994. [10J M.-Y. Jiang, Hofer-Zehnder symplectic capacity for 2-dimensional manifolds. Proc. Roy. Soc. Edinburgh (1993), 123A: 945-950.
A symplectic transformation and its applications
187
[11] M.-Y. Jiang, Periodic solutions of Hamiltonian systems on hypersurfaces in a torus, Manuscripta Math., 85(1994), 307-321. [12] M.-Y. Jiang, Symplectic embeddings from (R2n,wo) to some symplectic manifolds, Proc. Roy. Soc. Edinburgh, 129A, 1999( to appear) . [13] M.-Y. Jiang, Periodic motions of particles on torus with a magnetic field. Research Report No.21, Institute of Mathematics, Peking University, 1997. [14] M.-Y. Jiang, Periodic solutions of a class of Hamiltonian systems on the cotangent bundle of torus, Research Report No.27, Institute of Mathematics, Peking University, 1998. [15] V. V. Kozlov, Variational calculus in the large and classical mechanics. Russian Math. Surveys (1985), 40: 37-71. [16] F . Lalonde, Isotopy of symplectic ball, Gromov's radius and the structure of ruled symplectic 4-manifolds. Math. Ann. 300 (1994), 273-296. [17] F . Lalonde and D. Mc Duff, The geometry of symplectic energy. A nn. of Math . 141 (1995),349-371. [18] D. Mc Duff and D. Salamon, Introduction to Symplectic Topology, Oxford Mathematical Monographs: Clarendon Press, 1995. [19] G. C. Lu, The Weinstein conjecture on some symplectic manifolds containing the holomorphic spheres, Kyushu J. of Math., 52 (1998), 331-351. [20] S. P. Novikov, The Hamiltonian formalism and a multivalued analogue of Morse theory. Russian Math. Surveys 37(1982), 1-56. [21] S. P. Novikov and P. G. Grinevich, Nonselfintersecting magnetic orbits on the plane. Proof of the overthrowing of cycles principle. Amer. Math. Soc. Transl. (2) 170 (1995), 59-82. [22] P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math ., 31 (1978), 157-184. [23] P. H. Rabinowitz, Periodic solutions of large norm of Hamiltonian systems, J. Diff. Equa., 50 (1983),33-48. [24] P. H. Rabinowitz, Minimax Methods in Critical Point theory with Applications to Differential Equations, CBMS, Regional Conf. Ser. in Math., 65, Amer. Math. Soc., Providence, RI, 1986. [25] I. A. Taimanov, Closed extremals on two-dimensional manifolds. Russian Math. Surveys (1992), 47: 163-211.
Positive Solution to p-Laplacian Type Scalar Field Equation in RN with Nonlinearity Asymptotic to u p - l at Infinity 1 Gongbao L£2, Lina Wu and Huan-Song Zhou Young Scientist Laboratory of Mathematical Physics Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences P .O.Box 71010, Wuhan 430071, P .R. China Email: ligblilwipm . whcnc.ac . cnandhszhou~wipm.whcnc.ac.cn
Abstract
We consider the following elliptic problem:
-div(lV'ulp-2V'u) + mlul P - 2 u { u E W1,P(R N ), N > p > 1,
= f(x, u)
where m > 0, ,~f:.'..t tends to a positive constant as u -+ +00. In this case, f(x, u) does not satisfy the following Ambrosetti-Rabinowitz type condition, that is, for some () > 0,
Ai"
o :5 F(x, u) =
0
1 () f(x, u)u for all (x, u) E R N x R, f(x, s )ds :5 p +
which is important in applying Mountain Pass Theorem. By a variant version of Mountain Pass Theorem, we prove that there exists a positive solution to the present problem . Furthermore, if f(x, u) == f(u), the existence of a ground state to the above problem is also proved by using artificial constraint method.
1
Introduction
In this paper, we deal with the existence of a positive solution to the following non-autonomous quasilinear scalar field equation: -div(lV'ulp-2V'u) + mlul p - 2 u { u E W1,P(R N ), N > p > 1, 1 Partially Supported by NSFC 2Partially Supported by Academy of Finland
188
= f(x,u)
in RN,
(1.1)
Positive solution to p-Laplacian type scalar field
189
where m > 0 is a constant. The conditions imposed on f(x, t) are the following: (Cl): f: RN x R ~ R satisfies the Caratheodory conditions, i.e. for a.e. x E RN, f(x, t) is continuous in t E R and for t E R, f(x, t) is Lebesque measurable with respect to x ERN; f(x, t) ~ 0, "It ~ 0, x ERN; f(x, t) == 0, \:Ix E RN, t < O. (C2): lim ~1::P = 0 uniformly in x ERN . t--+o
(C3) : There is a constant f E (0, +00) such that lim ~1::~) = f uniformly t--++oo in x ERN. (C4) : For each x E RN, ~1::P is non decreasing with respect to t > O. RN
(C5) : There is a function f(t) E C(R) with f(x, t) ~ f(t) for any (x, t) E N : f(x, t) > !(t)} > 0 for any t > 0 such that X Rand mes{x E R lim f(x, t) = f(t) uniformly in bounded t. 1"'1--++00 (C6):
!
E Cl(R) and
(p - 1)!(t) < f'(t)t for all t > O.
Definition We say that u E W1,P(R N ) is a nontrivial (weak) solution to problem (1.1) if u t 0 and satisfies
r {1Y'ulp-2Y'u.Y'cp+mluIP-2ucp}dx = iRN r f(x, u)cpdx iRN
for all cp E W1,P(R N ).
Moreover, we say that u E W1,P(R N ) is a positive solution to problem (1.1) if u is a nontrivial weak solution of (1.1) and u(x) > 0 a.e. in RN . Clearly, solutions to (1.1) correspond to critical points of the following energy functional defined on W1,P(R N ), I(u) =
!
P
r (lY'uIP+mluIP)dx- iRN r F(x,u)dx, iRN
where F(x,u) =
1"
f(x,s)ds .
0
(1.2) The very useful tools to get critical points of I(u) are the famous "Mountain Pass Theorem" proposed by Ambrosetti and Rabinowtiz in their paper [1] and the constraint minimization. Combining Mountain Pass Theorem and Sobolev imbedding theorem, people have attained a lot of achievements in studying the existence of solutions of nonlinear Dirichlet problem in bounded domain n (see, e.g. [13]). Problem (1.1) occurs in many applications and there is an enormous literature concerning the existence of nontrivial solutions for problems similar to (1.1), or its bounded-domain counterpart under various hypotheses on f(x, t), see, e.g. [1] [2] [4] [11] [12] [18] [20] and the references therein.
Gongbao Li, Lina Wu and Huan-Song Zhou
190
Semi-linear elliptic equations in R N , e.g. problem (1.1) with p = 2, were motivated in particular by the search of certain kinds of solitary waves in nonlinear equations of the Klein-Gordon or Schrodinger type (see, e.g. [2) [3) [4)). Evidently, a contrast between semi-linear elliptic boundary value problems on a bounded domain and on RN is the apparent lack of compactness of Sobolev imbedding in treating latter. A partial reason for the lack of compactness of the Sobolev imbedding is the invariance of RN under the translations, rotations and dilations. To overcome this difficulty, people have attempted various methods, e.g. by working with some appropriate constraint in order to have some compactness. For example, to the following scalar field equation:
-D.u = f(x,u), { u E HI (RN),
x ERN, u"t. 0,
(1.3)
when f(x, u) is spherical symmetrical (i.e. f(x, u) = f(lxl, u)) or f(x, u) is autonomous (i.e. f(x,u) == f(u), "Ix ERN), we can consider problem (1.3) in the radial symmetrical Sobolev Space H~(RN) = {u E HI(R N ): u(x) = u(lxl), "Ix ERN}.
By the compactness of the imbedding H:(R N ) Y Lq(RN)(2 < q < ~~2 if N > 2), the existence of solutions to problem (1.3) in the Sobolev space H: (RN) were considered by Strauss[4]' Berestycki-Lions[2)[3). However, these methods does not work for general f(x, t). For this purpose, the "ConcentrationCompactness Principle" was proposed by P-L.Lions, in which some necessary and sufficient conditions for the compactness of minimizing sequence of a functional in unbounded domain are given. Based on this principle, the variational elliptic problems in RN were studied widely, see [6).....,[10) and the references therein. In [12], Yang and Zhu studied problem (1.1) under the basic assumptions similar to (C1) (C2) (C4)-(C6) and f(x, t) is subcritical, furthermore, they require as usual the following technical condition proposed by Ambrosetti and Rabinowitz in [1], that is, for some () > 0,
(U
1
0::; F(x,u)~ 10 f(x,s)ds::; p+(}f(x,u)u, V(x,u) ERN
X
R.
(1.4)
This condition implies that for some C > 0, F(x, u) 2: Clul P+9 for u > large, which makes the functional J satisfy the "mountain pass structure" and (1.4) also ensures that every (PS)c sequence {un} is bounded in HI (R N ). By applying the concentration-compactness principle, they proved that J(u) satisfies (PS)c condition for < c < Joo (see below for the definition of J oo ),
°
°
Positive solution to p- Laplacian type scalar field
191
hence the existence of a nontrivial solution of (1.1) is obtained by Mountain Pass Theorem. However, (C3) implies that !(x, t) is asymptotically "linear" with respect to u p - 1 at +00, and (1.4) can not be true. The study of the existence of positive solution to problem (1.1) under the condition (C3) is related to seeking special solutions of Maxwell's equations (see, e.g. [14] [15]). In this case, problem (1.1) with p = 2 have been studied in the spherical symmetric Sobolev space by Stuart and Zhou in [20] [21] [28]. Without assuming the symmetry on !(x, t), Jeanjean in his paper [18] proved that there is a positive solution to problem (1.1) with p = 2 under more or less the same assumptions as (C1) - (C4), moreover, he requires that !(x, t) is I-periodic in x ERN, which makes problem (1.1) invariant under suitable translations. But, he can't assure whether this solution achieved the mountain pass level of the energy functional / or not. Recently, Li and Zhou in [16] proved the existence of a positive solution to problem (1.1) for p = 2 under more general assumptions on !(x, t) as (C1)-(C6) with p = 2 and proved that this solution can achieve the mountain pass level of /. The aim of this paper is to generalize the main results of [16] for general p > 1. To state our main results, we first define the problem at infinity associated with problem (1.1) as follows: -div(lV'uIP:"'2V'u) { u E W1,P(R N ),
+ mlul p - 2 u =
f(u), x ERN,
(1.5)
and for any u E W1,P(R N ), define /OO(u) = where P(u)
~
p
= IoU f(s)ds.
[
iRN
(lV'uI P + mlulP)dx - [
iRN
P(u)dx,
(1.6)
Clearly, /00 E C1(W1,P(R N ), R). Denote
A = {u E W1,P(R N ) : (/00' (u),u)
= O,u ~ O},
(1.7)
where (', -) is the usual dual paring between W1,P(R N ) and its dual space W-1,p' (RN) (p' = f-r), [00' is the Frechet derivative of / 00 . It can be sho~n that A :f. 0 (see Lemma 3.1 in section 3 below). We define now the followmg minimization problem: JOO
= inf{IOO(u) :
u E A}.
(1.8)
We recall that if a solution of (1.5) achieves Joo in (1.8), it is called a "ground state" for (1.5) . Our main results are as follows:
Gongbao Li, Lina Wu and Huan-Song Zhou
192
Theorem 1.1. Let e E (m, +00) and conditions (C1)-(C6) hold, then Joo > 0 and it is achieved by some Uo E Wl'P(RN). Moreover, uo(x) > 0 a.e. in RN, which is a ground state for problem (1.5). Theorem 1.2. Suppose that conditions (C1)-(C6) hold, then problem (1.1) has a positive solution if E (m, +00) and there is no positive solution to (1.1) if e:s; m .
e
We mention that, to our best knowledge, the above results have not been seen elsewhere. To prove our main results, the difficulties are caused by the lose of condition (1.4) and the compactness of Sobolev imbedding in RN Since lim ~1~~} =
t-++oo
e < +00,
(1.4) is not satisfied, even in the case of p = 2, Joo > 0 is no longer apparent and not every minimizing sequence of Joo is apparently bounded as in the case of lex, t) satisfying (1.4). To prove Theorem 1.1, we use the Ekeland's variational principle on Finsler manifold (see Lemma 2.6 in Section 2, or [16]) to get a special minimizing sequence {un} of Joo in W1 ,P(R N ) with 11100 (un)llw-l.pl (RN) ~ 0, then as in [16] we use the concentration-compactness principle to show that Joo > 0 and Joo is achieved. To prove Theorem 1.2, as in [16] we find that the positive solution Uo of (1.5) obtained by Theorem 1.1 satisfies 1 (Uo ( I)) --+ - 00 as t --+ + 00 and we can construct the mountain pass level as follows c = inf max leu) , (1.9) 1
-yEr uE-y
where
r
=
b
E C([O, 1], Wl 'P(RN)) : 'Y(O) = 0, 'Y(1) = uo( ~)} with some
to to > 0 large enough. Then by the Mountain Pass Theorem without (PS) condition as in [19], there is a (PS)c sequence of 1 such that
To prove that {un} is bounded in W1,P(R N ) is a typical difficulty in the case of (C3) for the lack of (1.4). For this step, we follow basically the processes as in [16], of course, there are some technical difficulties while we work in the Banach space Wl ,P(R N ) with general p > 1 instead of the Hilbert space Hl(RN) = Wl,2(RN) . To overcome the difficulty of the lack of compactness of Sobolev imbedding from W1,P(R N ) to U(RN), we use a different approach from that of [16] . Instead of using the complete long procedure of concentration-compactness principle, we find that by comparing the energy level of two different critical points, some desirable results can be obtained. More precisely, we prove that
Positive solution to p-Laplacian type scalar field
193
°
if {Uk} is a bounded (PS)c sequence of [ with c > and Uk ~ Uo weakly in WI,P(R N ), then either Uo E WI ,P(R N ) \ {o} with I'(uo) = 0, or c ~ Joo. This is similar to Proposition 3.1 in [17), but we don't assume that f(x,') E CI as [17] did. In fact, the method of [17] seems only applicable to the case where m f(x, u) = L Qi(x)lul qi - 2 U (qi > p, Qi(X) E LOO(RN )). As we can show by i=1
using Pohozaev identity for p-Laplacian type equation (see Lemma 2.1 below) and the definition of c in (1.9) that c < J oo , then Theorem 1.2 is proved. Clearly, our approach here is much simpler than that of [16] even if p = 2. We end this section by giving some notations. Throughout this paper, we denote by C a universal positive constant unless specified and, define the norm of u E WI,P(R N ) by
lIuli ~ Ilullwl.P(RN) = (IN (IVuI P + m1uIP)dX)
l. P ,
and the norm of u E £p(RN)(1 < p < +00) by
We use" -t" (" -''') to indicate the strong (weak) convergence in corresponding function spaces.
2
Preliminary results
In this section, we give some preliminary results which will be used frequently in the following text. By (Cl), it is clear that f(x, t)
= f(x, t+),
V(x, t) E RN
X
R.
(2.1)
It follows from (Cl) - (C5) that for any c > 0, q E (p,p*), (N > p), there exists CE > such that for all x ERN, t ~ 0,
°
:!p
f(x, t) :::; ctp F(x, t) :::;
I
+ CEt q - l ,
clW + CEIW,
f(t):::; ctp F(t):::;
I
+ CEt q - l ,
p*
=
(2.2)
clW + CEIW,
(2.3)
f(x, t) :::; UP-I, f(t):::; Up-I.
(2.2')
By (C4) (C5) we see that F(x, t) < ~ f(x, t)t -p
for all x E RN, t
~ 0,
F(t):::;
~p f(t)t
for t
~ 0.
(2.4)
Gongbao Li, Lina Wu and Huan-Song Zhou
194
Combining (C2) (C3) and (C5), we have that lim f(t)
t-tO t p - 1
= 0,
lim
f(t)
t-t+oo tp-l
=t.
(2.5)
FUrthermore, (C6) implies that
f(t) tp -
is strictly increasing in t
1
> 0 and F(t) ~
Jot
f(s)ds <
!p f(t)t,
'tit> O. (2.6)
Lemma 2.1. (Pohozaev identity) Under conditions (C1)-(C6), if Uo is a weak solution of (1.5), then Uo satisfies a Pohozaev type identity:
(2.7)
Proof.
Let g(u)
= /(u) -
mlul p- 2 u, then Uo satisfies
-~p(uo) = g(uo), where ~pu = div(IVuI P- 2 Vu).
Hence, to show (2.7) we need only prove that
f
JRN
IVuolPdx = NNP
f
-pJRN
G(uo)dx, where G(t) = ft g(s)ds.
Jo
(2.7')
We turn now to showing (2.7)'. For any fixed R > 0, by the Pohozaev type identity for p-Laplacian in bounded domain BR in [25) and noticing that G(t) is independent of x, that is, aG",(t) == 0, so we have
l
BR
[NG(uo)
NIl
+ p--uog(uo))dx = (1-
P
-)
P
where v denotes the unit outward normal to aBR. On the other hand, it is not difficult to see that
Hence
8BR
lauo 8 I dS, P
(x, v)
V
Positive solution to p-Laplacian type scalar field
195
We claim that there exists a sequence Rn ---+ +00 such that
r
Indeed, if liminf R lV'uolPdS = Q > 0 and by J+oo i'fdR R--t+oo 18BR 1 oo that Jo+ (J8BR lV'uoIPdS)dR = +00 which is impossible, since
= +00
we see
Therefore
P-Nj -
P
and Since
N -~p(uo)
B~
j RN
n P-Nj uog(uo)dx ---+ --
P
~
uog(uo)dx,
P-Nj G(uo)dx + uog(uo)dx = P RN
= g(uo)
o.
and by integrating by parts we have
Lemma 2.2. Let {Pn} C Ll(RN) be a bounded sequence and Pn ~ 0, then there exists a subsequence, still denoted by {Pn}, such that one of the following two possibilities occurs: (i) (Vanishing): lim sup J, +B Pn(x)dx = 0 for all 0 < R < +00. n--t+oo yERN y R (ii) (Nonvanishing): There exist Q > 0, R < +00 and {yn} C RN such that lim n--t+oo
r
lYn+BR
Pn(x)dx
~ Q > O.
The proof of this lemma is trivial. Lemma 2.3. (Vanishing Lemma, Lions [9]) Let 1 < P ~ +00, 1 ~ q < +00 with q:f:. :!p if P < N . Assume that {un} is bounded in U(RN), {V'u n } is bounded in £p(RN) and sup J, +B lunlqdx ~ 0 for some R > 0, then
yERN y
R
196
Un ~
Gongbao Li, Lina Wu and Huan-Song Zhou
°
in U>:(RN) for any a between q and NNp .# -p
Lemma 2.4. For I defined by (1.2) , if {un} C Wl ,P(R N ) satisfies (1'(un),U n ) ~ 0. Then, for any t > 0, by extracting a suitable subsequence, we have
1(tun )
~
1 + tP pn
- - + 1(u n ).
°
In particular, if 'In ~ 1, (I' (un), un) = then 1(tun ) ~ 1(u n ) for any t > 0. Proof. This Lemma is proved in [20] for P = 2 and in [27] for general p> 1.# Finally, we give several results for the minimization problem (1.8) . For this purpose, we recall some results related to Finsler manifold. Let X be a real Banach Space. /, gl, g2 . .. gn : X --+ Rl be in C l ,
M = {x EX : gi(X) = O,i = 1,2", ·,n}, where {g~(x)}r are linearly independent for any x E M, then M is a submanifold of X with the natural Finsler structure (see [26]), hence it is a complete Finsler Manifold. Also, for any p E M there is a coordinate neighborhood in M which is diffeomorphic to {g~(p),gHp)"" ,g~(p)}.J.., i.e. there is the direct sum decomposition X = Xl ffispan{el,e2,· ·· ,e n } such that there is a neighborhood U of p in M such that U is diffeomorphic to a neighborhood V of () EX, where ei E X satisfies
and the tangent space Tp(M) of M at p satisfies Tp(M) we have the following lemmas: Lemma 2.5. (Lemma 2.4 of [16]) have
~
Xl' FUrthermore,
With the same notations as above, we
n
(d/IM(p),7rV) = (f'(p),v) -
I)!' (p),ej)(gj(p),v)
for all v E X,
j=l
where
11'
is the projection of x to Tp(M)
~
Xl.
The following result is the Ekeland's variational principle on Finsler manifold.
Positive solution to p-Laplacian type scalar field
197
Lemma 2.6. (Corollary 1.3 in Chapter 2 of [26]) Let M f:. > be a complete Finsler manifold, f E Cl(M,R) be bounded from below, then there is a sequence {Pn} C M such that
In particular, if f satisfies (PS) condition, then there is a Po E M such that f(Po)
= pEM inf f(p)
and df(po)
= 0.#
Now we study A defined by (1.7). By f(t) E C 1 (R), for any u E W 1 ,P(R N ) the functional defined by
is a C 1 functional and for any u E A, we have u+ for all s < 0 and u -:t. o. Hence
(g'(u),u)
r (lV'uI iRN
-:t. 0 by the facts of f(x, s) == 0
r [f(u)u + f'(u)u iRN
=
p
=
r [(p iRN
l)f(u)u - f'(u)u 2 Jdx
r [(p iRN
l)f(u+) - f'(u+)u+Ju+dx < 0,
P
+ mlulP)dx -
2
Jdx
by (C6).
So A is a closed hence complete submanifold of W 1 ,P(R N ) with the natural Finsler structure. Therefore, using Lemma 2.5 with el = (g'(~)'u) we have Lemma 2.7.
For any u E A, Vv E W 1 ,P(RN
(dJ OO IA(U),1TV)
),
(loo' (u),v) - (loo' (u),ed(g'(u),v)
=
(loo' (u), v)
(by
g(u)
= 0,
Vu E A),
where 1T is the projection from W 1 ,P(RN ) to TuA.# By Lemma 2.6, Lemma 2.7 and noting A f:. 0 (see Lemma 3.1 in Section 3), we have Lemma 2.8. There is a sequence {Un} C A such that
JOO(u n ) ~ Joo
= uEA inf JOO(u)
and
J oo ' (un) ~ 0 in W- 1 ,p' (R N ).#
198
Gongbao Li, Lina Wu and Huan-Song Zhou
3
Proof of Theorem 1.1 Before proving Theorem 1.1, we give two lemmas. Lemma 3.1. If (C1)-(C3) (C5) hold and l > m, then A -::f. Proof. We denote
For a
cp.
N
> 0, setting wa(x) = (d(N))-laP'I e-alxIP, it is easy to get that Iwa(x)lp
= 1,
l\7wa(x)l~
= aD(N)
and
then by (2.2), it is easy to see that
(I' (twa (x)), twa(x)) 2::
° for
On the other hand, if we choose a E (0,
t>
° small enough.
(3.1)
b(J:;)), where l is given by (C3), then (3.2)
and by Fatou's lemma
IlwaliP-
lim
{
t--t+oo ) RN
!(x, twa)twa dx tP
lim !(x, tWa~W~ dx tP-Iw~ 1
< IlwallP - {
) RN t--t+oo
= =
Ilwall P -llwal~ l\7wal~
+ (m -l)lwal~ < 0,
by(3.2).
So (I'(tw a ), tWa) -+ -00 as t -+ +00, this and (3.1) imply that there is at· > large enough such that (I' (t·w a ), t·w a ) = 0, t·w a E A.#
°
Proposition 3.1. (Theorem 5 in [22]) Let 0 be a smooth domain in RN (possibly unbounded) , u E C 1 (0) be such that ~pu E L~oc(O), u 2:: a.e. in O. ~pu S f3(u) a .e. in 0 with f3 : [0, +00] -+ R continuous, nondecreasing, f3(0) = and either f3(s) = for some s > or f3(s) > for all s > but 1 ' Jo1(j(s))-pds = 00 holds, where j(s) = J; f3(t)dt. Then if u does not vanish
°
°
°
°
°
°
199
Positive solution to p-Laplacian type scalar field
identically on fl, it is positive everywhere in fl .# Lemma 3.2. If (C1)-(C3) and (C6) hold, then Joo > O. Proof. If Joo = 0, by Lemma 2.8 there is a sequence {un} C A such that
(3.3)
First of all, we show that {un} is bounded in Wl'P(RN). By contradiction, we suppose that (3.6) lIu n ll -+ +00 as n -+ +00, and for any fixed a > 0, let
Clearly, {w n }, {w;i} are bounded in Wl,P(R N ). For Pn(x) = Iw;t(x)jP, the conditions of Lemma 2.2 are satisfied. We claim that neither (i) nor (ii) of Lemma 2.2 holds. Hence, (3.6) can not be true, that is, {Un} is bounded in W1 ,P(R N ). In fact, if there is a subsequence of {Pn}, still denoted by {Pn}, such that Lemma 2.2 holds. We get contradictions in both cases. Case 1
Vanishing: In this case, by Lemma 2.3 and (2.3), we see that
So
1 1 P P = -lIw n ll + 0(1) = -a + 0(1), p p
by(3.7)
(3.8)
where, and in what follows, we denote by 0(1) the quantity which tends to 0 as n -+ +00. But, by Lemma 2.4 and (3.3)
(3.9)
Gongbao Li, Lina Wu and Huan-Song Zhou
200
which is impossible by (3.8) . Case 2 Nonvanishing: In this case, there are 1] > 0, R RN such that lim ( Iw;t(x)IPdx 2: 1] > O.
> 0 and {Yn}
C
(3.10)
n--++oo lYn+BR
Set w;t(x) = w;t(x + Yn), then IIw;t(x)11 = Ilw;t(x)11 ~ IIwn(x)11 = a and by Sobolev imbedding we may assume that for some w(x) E Wl,P(R N )
w;t(x) ~w(x) weakly in W1,P(R N ), w;t(x) ~w(x) strongly in Lioc(R N ), w;t(x) ~w(x) a.e. in RN , these and (3.10) imply that
w(x)
't 0
and w(x) 2: 0 a.e. in RN.
(3.11)
By (3.6) (3.7) and (2.6), for n large enough, we have tn =
a
Ilunll
() l(tnu;t) !Cu;t) E 0,1 and (tnun)p-l + < + ' - (Un)p-l
hence
then it follows from (2.6) and Fatou's Lemma that
= p~lIwnliP >
( P(wn)dx 2: l [~l(w;t)w;t hN hN P
inN [~!Cw)w - pew)] dx + 0(1).
So, by (2.6) and (3.9) we have
P(w;t)] dx
Positive solution to p-Laplacian type scalar field
201
this means tV == 0 which contradicts (3.11). Thus, {un} is bounded in W1,P(R N ). Letting Pn(x) = lun(x)IP and by Sobolev imbedding that {Pn} is bounded in Ll(RN), then applying again Lemma 2.2 we know that for some subsequence of {Pn(x)} either Vanishing or Nonvanishing occurs.
Case I
If Vanishing occurs: Similar to (3.8), we have
1 [OO(u n ) = -lIunll P + 0(1).
(3.12)
P
Taking
€
= T in (2.2), it follows from Ilunll P = £N
this means that
(3.4) that
f(un)undx ::::;
1 211unilP : : ; Cllunll q
~llunllP + Cllunll q ,
(q > p). Hence, there is a fJ > 0, such that (3.13)
[00 (Un) ~ Joo
So, if
Case II such that
= 0, (3.12) and (3.13) are contradictory.
> 0, R > 0, {Yn}
If Nonvanishing occurs. There exist TJ lim n-too
( lun(x)IPdx iyn+BR
~ TJ > O.
C RN
(3.14)
Let un(x) ~Un(X + Yn}, we claim that for any cP E COO(RN),
{
iRN Indeed, so
[lV'u n IP- 2V'u n V'cp + mlun lp - 2u ncp] dx - (
Vcp E Wl'P(RN), let CPn = cp(x -
1([00 (Un), cp)1 1
f(un)cpdx
iRN
Yn}, it is easy to see that
=
I£N [lV'unIP- V'unV'cp +
=
1£)V'unIP-2V'un V'cpn
2
p
mlunl
-
(3.15)
IICPnl1 = Ilcpll,
2 uncp - f(un)cp)dxl
+ mlunlp - 2unCPn - f(un)CPn)dxl
1 1(I001 (un), CPn)1 : : ; 11[00 (un)IIIICPnll = 11[001(un)llllcpll ~ 0 by (3.5), and (3.15) is proved.
= 0(1).
Gongbao Ll~ Lina Wu and Huan-Song Zhou
202
Since {un} is bounded in W1,P(R N ), {Un} is also bounded in W1,P(R N ), then we may assume, by Sobolev imbedding, that for some u(x) E Wl,P(R N ),
~ u(x) weakly in W1,P(R N ), Un (x) ...2:.t u( x) strongly in Lioe (RN), { un(x) ...2:.t u(x) a.e. in R N , un(x)
(3.16)
and u(x) t. 0 by (3.14). By (3 .16) and (2.2), it is clear that for any t.p E W1,P(R N ),
(
iRN
[/(Un) - /(u)]t.p(x)dx...2:.t 0 and (
iRN
m[lu nlP- 2u n - luIP-2U]t.pdx...2:.t o.
On the other hand, by (3.16) and (3.5), we can show that for any t.p E W1,P(R N ), { [lVu IP - 2 Vu - IVuIP- 2Vu]Vt.pdx...2:.t 0,
iRN
n
n
which is trivial if p = 2, but for general p compactness principle to show first that
> 1 we need to use the concentration(3.17)
(see the proof of Theorem 1.6 in [23] for details) and then the result follows. So U is a weak solution of -Llpu + mlulp-2u = /(u), that is,
{
iRN
[lVuI P- 2VuVt.p+mlul p- 2ut.p]dx- {
iRN
/(u)t.pdx = 0, for any t.p E Wl'P(RN). (3.17')
Taking t.p = u - in the above formula, it is easy to see that u - == 0 by noticing (Cl), hence u ~ 0 a.e. in RN and by standard regularity results [29] u E CI~~(RN) . Using Proposition 3.1 with {3(u) = mlul p - 1 , and noticing Llp(u) = mlul p- 2u - /(u) ~ {3(u), we see that u is positive everywhere in RN. Then using again (2.6) and Fatou's lemma, we have
which means that Nonvanishing is also impossible. So, Joo > 0.# Proof of Theorem 1.1. By Lemma 3.2, to prove Theorem 1.1, it is enough to show that Joo is achieved by some Uo E W1,P(R N ) and Uo > 0 a.e. in RN.
Positive solution to p-Laplacia.n type scalar field
203
Just like in the proof of Lemma 3.2, it follows from Lemma 2.8 that there exists {un} C W1 ,P(R N ) such that (3.3)-(3.5) hold, and keep in mind that Joo > 0 from now on. Step 1 {un} is bounded in Wl ,P(R N ). If
lIunll ~ +00, we let
Set Pn(x) = Iwn(x)IP, ifthere is a subsequence, still denoted by Pn(x), such that Lemma 2.2 holds, then by the same processes as in getting (3.8) (3.9), and noticing that (3.19) we know that Vanishing doesn't occur. If Nonvanishing occurs, i.e. there are {Yn} C RN, R
Wn(X) ~ wn(x
weakly in Wl,P(RN),
wn(x) ~ w(x)
strongly in Lroc(RN) p::; q
wn(x) ~ w(x)
a.e. in RN
= un(x + Yn),
> 0 such that
=!. 0 satisfying
Wn(x) ~w(x)
Set un(x) we get:
'1/
+ Yn), Ilwn(x)11 = Ilwnll = p(Joo ) t,
hence , there is some 'Ill E W1,P(R N ) and 'Ill
{
> 0,
then wn(x)
= ilfi~(m,
< p.
= :!!p,
(3.20)
and 'VIP E C(f(RN) by (3.15)
that is, 'VIP E C(f(RN),
r [I'VwnIP-2'Vwn'VIP + mlwnl iRN
p
where
Pn ( X ) =
-
2w IP]dx n
llid _p_l , Un
{ 0,
r
iRN
Pn(x)(w;t)P-1IPdx
= 0(1), (3.21)
204
Gongbao Li, Lina Wu and Huan-Song Zhou
Similar to (3.17) we have \7w n ~ \7w a.e. in RN, then Vtp E Co (RN) by (3.20) (3.21)
{ [I\7wI iRN
P - 2 \7w\7tp+
iRN Pn(x)(w~)P-ltpdx+o(l).
mlwl p- 2wtp)dx = (
(3.22)
By (2.5) (2.6), 0 :::; Pn(x) :::; £, {Pn(X)} C LOO(RN) and there is a hex) E LOO(RN) with 0 :::; hex) :::; £ a.e. in RN such that Pn(x).3. hex) weakly· in LOO(R N ). Since wn ~ w strongly in Lroc(RN), we get ...l!...-
Pn(X)(W~)p-l .3. h(x)(w+)p-l weakly in Ll~~' (RN),
{
iRN
Pn(x)(W~)P-ltpdx ~
(
iRN
h(x)(W+)p-ltpdx, Vtp E CO(R N ).
Thus by (3.22) and the density of Co(RN) in W1 ,P(R N ) we have for all tp E W1 ,P(R N ) that
{
iRN
[I\7uijP-2\7w\7tp
+ mlwl p- 2wtp)dx =
(h(x)(W+)P-ltpdx,
iRN
(3 .25)
taking tp = w- in the above formula, it is easy to see that w- == O. So, w a.e. in RN and w is a weak solution of
~
0
Applying Proposition 3.1 as in the end of the proof of Lemma 3.2, we see ~
> 0 a.e. in RN. Noticing wn(x) = p(I~:iI Un ~ w(x) > 0 a.e in RN, then un(x) ~ +00 a.e. in RN and Pn(x) = ~~"::( ~ £, that is, hex) == £ Un
that w(x)
a.e. in RN and (3.25) implies that
{
iRN
l\7wl p- 2\7w\7tpdx
= (£ -
m) (
iRN
IwIP-1tpdx,
which is impossible since there is no nontrivial solution to -~pu = in W1,P(R N ) for any>. E Rl by Lemma 2.1, and Step 1 is proved.
>'lulp-2 U
Step 2 Joo is achieved by some Uo E Wl,P(R N ) and Uo > O. Let Pn(x) = lun(x)IP, by extracting a subsequence, we may assume that {Pn} satisfies Lemma 2.2. If Vanishing occurs, then by using Lemma 2.3 and (2.2),(2.3), we have
Positive solution to p-Laplacian type scalar field
205
Again by (3.4), we get
[OO(u n )
=
IN [tf(Un)Un - F(un)1dx ~
0,
which contradicts (3.3) since we have now Joo > O. So only Nonvanishing occurs. Then following the same procedures as in (3.14)-(3.17)1, there exists u(x) E WI ,P(R N ) with u(x) t= 0 such that
r [lVu(x)I JRN
P 2 - VuV
+ mlulp-2u
r f(u(x))
V
taking 0 a.e. in RN . Hence, u(x) E A and [OO(u) ~ Joo > O. On the other hand, it follows from (3.4) that 1 (l001(un) , un) = ([00 (un), un) = 0, that is,
Ilunll P =
r f( un)undx.
JRN
and
Hence,
4
u > 0 and [OO(u) = Joo
and Theorem 1.1 is proved.#
Proof of Theorem 1.2
In this section, we will use the following version of Mountain Pass Theorem to prove Theorem 1.2.
Proposition 4.1. ([19], Theorem I) Let E be a real Banach space with its dual space E* and suppose that [ E C I (E, R) satisfies the condition
max{I(O),I(ud} for some by
a < (3,
p
> 0 and UI
~
a < (3
E E with
~
inf [(u), lIull=p
IluII! > p.
c = inf max [(-y(t)), -yEro
Let
c ~ (3 be characterized
206
Gongbao Li, Lina Wu and Huan-Song Zhou
where r = b E C([O, 1), E): ')'(0) = 0, ')'(1) = ud is the set of continuous paths joining 0 and Ul. Then there exists a sequence {Un} C E such that
For the functional defined by (1.2), we have the following lemmas. Lemma 4.1. such that
If (C1)-(C3) and (C5) hold, then there exist p
l(u) ~ f3 > 0 for all
U
E W1'''(R N ) with
> 0, f3 > 0
lIull = p.
Furthermore, there exists Ul E W1 '''(R N ) with lIudl > p such that l(ud < O. Proof. Taking c = T in (2.3) and by using Sobolev imbedding, it is easy to see that the first part of this lemma is true. Now, let uo(x) > 0 be the solution of problem (1.5) obtained by Theorem 1.1, then loo(uo) = Joo > O. (4.1) For any fixed a
> 0, setting ua(x)
X = uo(-), a
(4.2)
then
l\7ual~
= aN-"I\7uol~,
IUal~
= aNluol~·
(4.3)
So by the definition of 100 given by (1.6), we have
(4.4)
Since N > p, we can find ao > 0 large enough such that
Hence, J(u ao ) < loo(u ao ) < 0 by (C5), and Lemma 4.1 is proved for Ul Lemma 4.2.
= u ao '#
Under the same conditions and u ao as Lemma 4.1, define c = inf max l(')'(t)) , "(HtE[O,l]
(4.5)
Positive solution to p-Laplacian type scalar field
207
where r = bE C([O, 1], Wl,P(R N )) : ')'(0) = 0, ')'(1) = uo: o }, then c E (0, JOO) . Proof. By Lemma 4.1, c ~ f3 > O. Let ')'(t) = uO(o:~t) for t E [0,1], it is easy to see that 1I')'(t)1I -+ 0 as t -+ 0, then we may set ')'(0) = 0 and ')'(t) is a continuous curve joining 0 and uo: o ' By (4.5) and the continuity of I(u), there exists E (0,1] such that
t*
=
c ~ sup I(')'(t)) tE[O,l]
=J
(UO(o::t*))
x sup I(uo(-)) O:ot
tE[O,l]
<1
00
(UO(o::t*))'
by (C5).
(4.6)
Let Ut = Uo ( t), we claim that
Indeed, similar to (4.4) by Pohozaev identity (2.7) we have
(t N-
P
N-PN) -- - -t P Np
I 00 (ut} =
lV'uolPP
and it is easy to see that
d d/OO(Ut)
=0
if and only if t
= 1,
this implies (4.7), hence it follows from (4.6) and (4.7) that c < J oo .# Finally, we give the proof of Theorem 1.2. Proof of Theorem 1.2. Using (C3) (C4), it is clear that e ~ m if problem (1.1) has a positive solution. So, from now on, we suppose that eE (m, +00). By Lemma 4.1, for c defined by (4.5), it follows from Proposition 4.1 that there exists a sequence {Un} C Wl ,P(R N ) such that (4.8)
(1
+ Ilunll)ll/'(un)llw-l,P' (RN)
~ O.
(4.9)
Moreover, by (4.9) we have
(/'(u n ), un) ~ 0, (J'(un) , cp) ~O,
Vcp E W1,P(R N ).
(4.10) (4.11)
208
Gongbao Li, Lina Wu and Huan-Song Zhou
In what follows, we divide our proof into two steps. Step 1 {Un} is bounded in W1,P(R N ), hence there is some Uo E W1,P(R N) and a subsequence of {un}, still denoted by {Un}, such that
Un ~ Uo a.e. in RN. Just replace Joo by c in (3.18) and (3.19), following exactly the same procedures as Step 1 in the proof of Theorem 1.1, we know that if Ilunll....!:.t +00, by Lemma 2.2, Vanishing can not happen. If Nonvanishing occurs, then there are {Yn} C RN, TJ > 0, R > 0 such that lim n-too
Let wn(x) = wn(x + Yn), with W t; 0 such that
r
lYn+BR
Iwn(x)IPdx
Ilwn(x)11 = Ilwnll,
~ TJ > O.
then for some w(x) E Wl,P(R N)
(4.12) (4.13) Wn(x)
....!:.t w(x)
a.e. in RN.
(4.14)
Let un(x) = un(x + Yn) and for any cP E W1,P(R N ) setting CPn(x) then IICPnl1 = Ilcpll, similar to (3.15) we have
I(I'(un),CPn)1
= cp(x -
< III'(un)llw-l ,P'(RN)IICPnll
=
III'(un)llw-l.P'(RN)llcpll -t 0
hence
that is, for any cP E W1 ,P(R N
)
we have
as
n -t +00,
Yn),
Positive solution to p-Laplacian type scalar field
_j
RN
I(x + Yn, un) ()P-I d -p-l Wn cp X Un
209
0
(1)
i.e.
where I(x Pn (X ) =
{
+ Yn, un) -p-l' Un
0,
if Un> 0, if Un:::; O.
By (C3) (C4), we know that 0 :::; Pn{x) :::; l, then, by extracting subsequence, there exists some hex) : 0:::; hex) :::; l such that Pn(x).2l hex) weakly· in Loo.(O). Noticing . LPIDe (RN) ' strongIy In .J!...-p-l n -p-l strongly in LI~~l (R N ), wn -+ W -p-l n -p-l a.e. in RN . wn -+ W
-+
n Wn W-
Hence Pn{x)( w;t)p-l .2l h{x) (W+)p-l in L;I!::r (RN) and similar to (3.17), V'wn ~ V'w a.e. in R N , then
Thus
r [lV'wI P- 2V'wV'cp+mlwl p- 2wcpjdx r h(x)(W+)V-Icpdx, 'Vcp =
iRN
E WI,V(R
N ),
iRN
taking cp = w- , it is easy to see that w- == O. So W ~ 0 a.e in R N , that is, w is a weak solution of -div(lV'ulp- 2 V'u) + mlulp-2 u = h(x)(u+)P-l, by Proposition
210
Gongbao Li, Lina Wu and Huan-Song Zhou
3.1 and 6 p (u) a.e. in RN. Since wn N R , hence
= mlulp-2u 1.
= PIILi!
Pn () X
h(x)(U+)P-l ~ f3(u) ~mluIP-l, we see that W > 0
~ W > 0 a.e. in R N , we have that Un ~
Yn, un) = f(x +Un -p-l
+00
a.e in
n n n --+ (., that is, h(x) == (. a.e. in
So we get
which is impossible since there is no nontrivial solution for -6 p u = Alulp-2u for any A E Rl by (2.7), then {un} is bounded in Wl,P(R N ) and it is easy to see that Step 1 is proved by extracting a subsequence of {un} and some Uo E W1,P(R N ).
By Step 1, (2.2) (2.3) and the main results of [23] we have that V'U n ~ V'uo a.e. in RN and I'(uo)
= o.
It remains to show that Uo E Wl,P(R N) \ {O} . By contradiction, assume that Uo == O. If Un ~ 0 strongly in Wl ,P(RN), then lim [(un) = 0 which contradicts
lim [(un) = C > O. So Un n ..... oo that for some TJ > 0
n ..... oo
f+ 0 in Wl,P(RN) as
n ~
00
and we may assume
To finish the proof of Step 2, we need the following lemma: Lemma 4.3. Let (Cl)-(C6) hold, if {Uk} such that Uk ..!s. 0 in Wl ,P(R N ), then
Proof of Lemma 4.3.
c
Wl ,P(R N ) is a sequence
For any given R > 0 we see that
211
Positive solution to p- Laplacian type scalar field
~ I~ + I~ Since Uk .!:.. 0 weakly in W1,P(R N ) , by Sobolev's imbedding we know that Uk ~ 0 strongly in Lfoc(RN) for some subsequence of {Uk} (still denoted by {ud) and q is given as in Step 1. Hence (2.3) implies that lim I~ =
k-too
o.
On the other hand, for any given 8 > 0,
I~
[!
=
Izl>R
{lu~I<6}
+!
I.I>R {6:5IUkI9-'}
+!
I-I>R
{lu~li6-1}
jlF(X' Uk) -
P(uk)ldx
< cl(8) {
( IUkIPdx+C3(8) ( IUklpodx iRN IUkIPdx+C2(R)8-P iRN iRN
where cl(8)
=
IF(x'~~I~ F(t)1
sup
- t 0 as 8 -+ 0+, by (C2) ,
{~~I~:5; }
c2(R)
=
IF(x, t) - P(t)1 - t 0 as R - t +00 for fixed 8, by (C5),
sup I·I>R } { 6:5':5-6- 1
C3(8)
=
IF(x,
sup {
1-lo::R }
'0:: 6 -
1
Therefore, lim I~ k-too
?I; P(t)1 - t 0 as 8 -+ 0+ , p. = ;~ , by (C3). t p
= 0 and hence
Similarly, Lemma 4.3 is proved. Now we turn to the proof of Step 2. By Lemma 4.3 we have
c = I(Uk)
+ o~) = IOO(Uk) + 0(1)
{ (Ioo'(Uk),Uk) =J.Lk
= 0(1).
(4.16)
Gongbao Li, Lina Wu and Huan-Song Zhou
212
Let Uk(X)
= Uk(tkX),
where tk > 0 will be determined later, then (4.17)
(1 00 ' (Uk), Uk)
= t;;N (t~ - l)JRN IV'UklPdx = t;;N[(t~ - 1) JRN
+ t;;N (JOO' (Uk), Uk) IV'UklPdx + ILk] .
(4.18)
We claim that there is a A > 0 such that (4.19)
In fact, if (4.19) is false, then this implies that
JrRN IV'UklPdx ~ 0,
r
mluklPdx
Noticing !(X,Uk)Uk :S ~IUkIP
+ CIUkIP·,
JRN
but lim k--+oo
IIUkll P = rl',
~ fJP > O. we have that
this contradiction shows that (4.19) is true. Therefore we can set tk
=
(1 + JRN iV':k
1
IPdx ) P
such that Uk E A and
tk ~ 1. This facts together with (4.16) (4.17) imply that
which contradicts the fact that c < Joo (see Lemma 4.2). We have thus proved that Uo E WI ,P(R N ) \ {O} . Finally, by the same procedures as in the end of the proof of Lemma 3.2, we see that Uo > 0 in RN by Proposition 3.1. The proof of Theorem 1.1 is complete.#
Remark 4.1. We mention that under the assumptions (Cl)-(C6) we could actually show that Uk ~ Uo in WI ,P(R N ) and l(uo) = c by using the concentration-compactness principle as in [16] .
Positive solution to p-Laplacian type scalar field
213
References
1 A. Ambrossetti and R. Rabinowitz, Dual variational method in critical point theory and applications, J. Funct. Anal, 14(1973), 349-38l. 2 H. Berestycki and P.L. Lions, Nonlinear Scalar Field Equations, I: Existence of a Ground State, Arch. Rat. Mech.Anal., 82(1983), 316-338. 3 H. Berestycki and P.L. Lions, Nonlinear Scaler Field Equations, II: Existence of Infinitely Many Solutions, Arch. Rat. Mech . Anal., 82(1983), 347-369. 4 W.A. Strauss, Existence of solitary waves in higher dimensions, Comm. in Math. Phys., 55(1977), 149-162. 5 N. Berger and M. Schechter, Embedding theorems and quasilinear elliptic boundary value problems for unbounded domains, Trans. Amer. Math. Soc., 172(1972), 261-278. 6 P.L.Lions Principale de concentration-compacite en calcul des variations, C. R. Acad. Sci. Paris, 294(1982), 261-264. 7 P.L. Lions, La methods de concentration-compacite en calcul des variations. In Seminaire Goulaouio-Neter-Sohwarlz, 1982-1983, Ecole Polytechnique, Palaiseau, 1983. 8 P.L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, Part 1, Ann. 1. H. P. Anal. Nonli., 1(1984),109-145. 9 P.L. Lions, The concentration-compactness principle in the calculus of variations, The locally compact case, Part 2, Ann. 1. H. P. Anal. Nonli., 1(1984)4, 223-283. 10 P.L. Lions, The concentration-compactness principle in the calculus of variations, The limit case, part 1, Revista Matematica Iberoamericana, 1(1985)2, 45-120. 11 C.A. Stuart, Bifurcation for Dirichlet Problems without eigenvalues, Pmc. London Math. Soc., 45(1982), 149-162. 12 J.F. Yang and X.P. Zhu, On the existence of nontrivial solution of a quasilinear elliptic boundary value problem for unbounded domains, (I) positive mass case, Acta Math. Sci., 7(1987), 341-359; (II) The zero mass case, ibid, 8(1988),447-459. 13 P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. in Math., No. 65, AMS, Providence, R.1.l986. 14 C.A. Stuart, Self-trapping of an electromagnetic field and bifurcation from the essential spectrum, Arch. Rat. Mech. Anal., 113(1991), 65-96.
214
Gongbao Li, Lina Wu and Huan-Song Zhou
15 C.A. Stuart, Magnetic field wave equations for TM-modes in nonlinear optical waveguides, Reaction Diffusion Systems, Edit, Caristi and Mitidieri, Marcel Dekker, 1997. 16 G.B. Li and H.S. Zhou, The Existence of a Positive solution to asymptotically linear scalar field equations, Pmc. Royal Soc. Edinburgh, Section A, 130A(2000). 17 D.M. Cao, G.B. Li and H.S. Zhou, The Existence of two solutions to quasilinear elliptic equations on R N , Chinese J. Contemporary Math., 17(1966), 277-285. 18 L. Jeanjean, On the existence of boundary Palais-Smale sequences and application to a Landesmann-Lazer type problem, Pmc. Royal Soc. Edin., Section A, 129A(1999), 787-809. 19 D.G. Costa and O.H. Miyagaki, Nontrivial Solutions for Perturbations of the p-Laplacian on unbounded domains, J. Math. Anal. Applications, 193(1995),737-755. 20 C.A. Stuart and H.S. Zhou, Applying the mountain pass theorem to an asymptotically linear elliptic equation on R N , Comm. in P. D. E., 24(1999), 1371-1785. 21 H.S. Zhou, Positive solution for a semilinear elliptic equation which is almost linear at infinity, Z. angew. Math. Phys, 49(1998), 896-906. 22 J.L. Va'zquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12(1984), 191-202. 23 G.B. Li, The existence of a weak solution of quasilinear elliptic equation with critical sobolev exponent on unbounded domains, Acta. Math. Sci., 14(1994)1,64-74. 24 H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence offunctionals, Proc. Amer. Math. Soc., 88(1983),486-490. 25 M. Guedda and L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal. TMA, 13(1989), 879-902. 26 K.C. Chang, Critical Point Theory and Its Applications, Shanghai Scientific and Technical Press, 1986, (in Chinese) . 27 G.B. Li and H.S. Zhou, Asymptotically "linear" Dirichlet problem for pLaplacian, to appear in Nonli. Anal. 28 C.A. Stuart and H.S. Zhou, A variational problem related to self-focusing of an electro-magnetic field, Math. Meth. Appl. Sci., 19(1996), 1397-1407. 29 P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Diff. Eq., 51(1984), 126-150.
The existence and convergence of heat flows * Jiayu Li t
1
Introd uction
Let M and N be two compact Riemannian manifolds with dim M = m and dim N = n. In this paper we consider the existence and convergence of heat flows for harmonic maps. It was shown in [8] that the heat equation has a unique local solution, and that if the sectional curvature of N is nonpositive, it has a global solution and the solution converges to a harmonic map as t -+ 00. However (see [5], [7], [3], and [2]), in general the solution may blow up at a finite time or at infinity. Recently Lin-Wang [11] considered the exis~ence and the regularity property for the weak heat flow. They proved that if N does not carry a harmonic sphere S2, for any Uo E Hl,2(M, N), there is a stationary weak heat flow with initial data uo. We will give a new proof here which depends on the ideas developed in [13] . Furthermore, we show that the stationary weak heat flow subconverges to a stationary harmonic map. The main result can be stated in the following theorem.
Theorem 1.1 If the target manifold N does not carry any harmonic spheres S2, for any initial data Uo E Hl,2(M, N), there is a stationary weak heat flow u(x, t) with initial data uo, and u(x, t) subconverges to a stationary harmonic map u(x) as t -+ 00 . "This work is supported by NSF grant . tlnstitute of Mathematics, Academia Sinica, Beijing 100080, P. R. China.
215
Jiayu Li
216
For simplicity in notation, we assume in this paper that the metric on M is flat, i.e. we assume that M = Rm.
2
Strong convergence of the stationary weak heat flows
We recall the definition of a stationary weak heat flow, the two monotonicity inequalities (see [13]), and The to-regularity.
Definition 2.1 We say that u(x, t) E H 1,2(Rm x R+, N) is a stationary weak heat flow, if it is a weak heat flow and for almost all t > 0, (1) for any smooth vector field X on Rm with compact support,
(2) for any
dt
>
E Co(Rm) with >
j .
= -2
Rm
j' Rm
2': 0,
au )2dV - 2
1 Rm
au \l
It is clear that if u is a smooth heat flow, it is a stationary weak heat flow. Let u(x, t) E H 1 ,2(Rm x R+, N) be a stationary weak heat flow. Let q>(u, (x, to), r) be the functional defined in [15] .
q>(u, (x, to), r) = r2 {
I \l uI 2(y, t)G(x,to)(Y, t)dy
} S(r)
where S(r)
= Rm
x {t
= to - r2},
G(x,to)(Y, t) = (47f(to - t) )-~ exp( is the backward heat kernel.
'r - xl;)
4 to - t
The existence and convergence of heat flows
217
Lemma 2.2 Let u(x, t) E Hl,2(Rm x R+, N) be a stationary weak heat flow. Then for any 0 < rl < r2 :::; vItO, we have ~(u,
(x, to), r2) - ~(u, (x, to), rl)
> 2 -
i
to - r21 1
to-r~
Rm (to
aU
'Vu (Y - X ) 2 2(to _ t) ) G(x,to)(Y, t)dydt.
- t)( at -
We set
E(u, (x, t), r) = r 2- m
( Br(x)
J
I 'V u(y, t)1 2 dy
and
J(u, (x, t), r) = T2- m (
JBr(X)
lat ul 2dy.
Lemma 2.3 Suppose that J(u, (x, t), r) :::; T- 2a for 0 < r < TO, where 0 < a < 1. Then faT any 0 < rl < r2 < TO,
The Eo-regularity for smooth heat flows was proved by Struwe [15]. The partial regularity for stationary weak heat flows into spheres has been studied by Chen-Li-Lin [6] and Feldman [9], for the flows into a general target, the same result was proved by Liu [12] using the ideas in [6] and [1]. The theorem is as follows:
Theorem 2.4 Let u(x, t) be a stationary weak heat flow. Set
I 1 t r2
1 P(u, (x, t), r) = ~ r
There exists
EO
+
t-r2
Br(x)
> 0, such that if P(u, (x, t), r) <
EO,
I 'V ul 2dVdt.
Jiayu Li
218
then U is smooth in B i (x) x (t -
I 'V ul
sup Bi(x) x (t_
r2 4
r; ,t + r;) and ~ Cr- 2
,t+ r; )
where C > 0 depends on m, N, EO, E(u(·, 0)). Suppose that Uk is a sequence of stationary weak heat flows for harmonic maps with
By (2) in Definition 2.1, we have
(3) So we may assume that Uk --t U weakly in H I ,2(Rm x [0,00), N) . We set Et = nr>o{x E RmlliminfP(uk, (x , t),r) k--too
2:
EO}
to be the blow up set for the sequence Uk at t, and set E = Uo
= {t E (0, 00)1 liminf r (OtUk)2(X,t)dx = oo}, k--too } Rm
Bl = {t E (0, oo)llimf.l(E {--to
n (t - E, t + E)) > ~} , l
The existence and convergence of heat flows
219
and
A2
= U1B1,
A
= Al UA2 .
We also set, for any 0 < a < 1,
= nn U.l>r>O {x
So(t)
n
for any
E Rmllim inf I( Uk, k-HlO
(x, t), r) 2: r- 20 },
> 0,
€
Lemma 2.5 The measure of A , IAI is O. For any t E (0, (0) , we have Hm-2('L}) < +oo.Ift f/. A, and 0 < a < 1, Hm-2(1+o)(So(t)) < +00, and Hm-2(S;(t)) = 0 for any f > O. In this section, we mainly prove the following theorem which was first proved by Lin-Wang ([11]). We will use the ideas developed in [13] to give a new proof.
Theorem 2.6 Let Uk be a sequence of stationary weak heat flows for harmonic maps from Rm to N with energies E(Uk(', 0)) ::; A. If the target manifold N does not carry any harmonic spheres S2, there is a subsequence of Uk which we also denote by Uk such that Uk --+ U in Hl~;(Rm x R+, N) . And U is a stationary weak heat flow. Proof: It is clear th,at we may assume that Uk --+ U weakly in Hl ,2(Rm x R+, N) . The following lemma is a monotonicity inequality for the measure l/t ([13]). Lemma 2.7 Ifliminfk-+oeJ(ukl(x,t),r) ::; r- 2o , for 0 < r < ro, then for any 0 < rl < r2 < ro, we have
r~-m
r JB
I \l uI 2dV + r~-ml/t(Br2(X))
T2 (X)
~ r~-m
r
} BTl
I \l ul 2dV + r~-ml/t(Brl (x)) - c(r~-O - d- o ), (x)
where c > 0 depends on m, A and a
Jiayu Li
220
We proved the rectifiability for Et in [13]. Theorem 2.8 Let Uk be a sequence of stationary weak heat flows for harmonic maps from Rm to N with energies E(Uk(', 0)) ~ A. Let Et be the blow up set at t > O. Then it is a closed set. If t ~ A, Et is a rectifiable set. It is not difficult to see by Lemma 2.7 that, if t ~ A, Vt = O(x, t)Hm- 2l Et) where O(x, t) is upper semi-continuous in Et \ ScAt) with C(EO) ~ O(x, t) ~ C(A) for Hm-2_a.e. x E E t , C(EO) is a positive constant depending only on m and EO, C(A) is a positive constant depending only on m and A. So, O(x, t) is Hm-2 approximate continuous for Hm-2_a.e. x E Et. In the sequel, we assume that Uk does not converge strongly in Hl~; (Rm x R+) N), in other words, there exists t > 0 such that Vt(E t ) > O. We will show that we may derive a harmonic sphere S2 using the blow up analysis. We set B = {t E R+IVt(Et) > O} . By Corollary 6.2 in [13] we have B = (0, T) where T < 00, or T = 00 . The following Geometric Lemma was proved in [13]. Lemma 2.9 Suppose that x E Et \ Sn(t) and O(x, t) is Hm-2 approximate continuous at x. Then there exists r x > 0, such that, for any 0 < r < r x , we can find m - 2 points Xl, ... , Xm-2 in Br(x) n (Et\Sn(t)) so that, (1) IO(xj, t) -O(x, t)1 < Er , for j = 1"", m-2, where Er -+ 0 as r -+ 0; and (2) IXII 2': Smr, dist(xk, X + Vk-d ~ Sm r , for k = 1" . " m - 2, where Vk- l is the linear space spanned by {Xl - X,"', Xk-l - x}. Here Sm > 0 depends only on m . Using the Geometric lemma, we show the following lemma in [13] . Lemma 2.10 Let T E TEt . 1ft fj. A, we have lim lim f-+O k-+oo
JrB.(Et) I \iT ukl2dy =
O.
Here and in the sequel we denote by Bf(X) = {x E Rm I dist(x, X) < E}.
The existence and convergence of heat flows
221
Set Fk€(x)
=
r I VT ukI 2(x, x')dx' JB~(O)
for x E Et. Here and in the sequel, we denote by B;(x) the metric ball centered at x with radius r in R2. We consider the HardyLittlewood maximal function MFkf (x) of FkE{X)' which is defined by MFkf{X)
=
sup r 2 -
r
m
O
JB;:n-2(X)
FkE{X)dx .
Here and in the sequel we denote by B;.n-2(X) the metric ball centered at x with radius r in Rm-2. By the weak type (1,1) inequality for MFkf(X), we have
for any). > O. By Lemma 2.10, we have lim lim H m - 2 {x E Et E-+O k-+oo
I MFkf(X)
~).} = 0,
thus for any 1 > 0, H
m
-
2 (U:=1 n~=no Urc;=l n~ko {x
E EtIMFk (!,) (x)
1
~ l}) = o.
By the partial regularity result (cf. [6], [9], [12]), we can find t E R+ and Xl E Et \ (Sdt) U S;(t)) C B~-2(O), such that for any no > 0 and any ko > 0 there are nl > no and kl > ko satisfying 1
MFk,( "i) 1 (Xl) < -l.
(4)
and Uk(X, t) is smooth near (Xl, X') for all x' E B?(O), and (Xl, X')
rt
Sdt). 2 We claim that , for all k sufficiently large, we may find Ok --+ 0 such that max Ok2-m x/EBf(O)
1
B;:'-2(Xk)XB~k (x')
I V Uk 12( X, X')d XdX
I
= 8. to22m .
(5)
Jiayu Li
222
In fact, since Uk(X, t) is smooth at (Xk' x'), for any given k and for 0 < o(k), we have
If x E Et \ S;(t), we have
lim E( Uk, (x, t), r) ;:::
k-too
1 r -2 lim inf P( Uk, (x, t), -) 4 . 2m k-too 2
C€ -
c-/i.
So, for fixed 0 > 0 and sufficiently large k,
t
o2-m
} B;-2(Xk) x Bg(0)
I \l ukI2(X, x')dxdx';:::
to . 4· 22m
Therefore we can choose Ok > 0 so that (5) holds. By (4) and (5), since (Xk' x') f{. S!(t), we can find Ek --+ 0, rk --+ 0, 2 (Xk' xD E Et and a subsequence of Uk, which we also denote by Uk for simplicity, such that rk- m )
rB~-2(Xk) X B;k(xU I \l ukI 2(x, x')dxdx' = 8· to22m
The existence and convergence of heat flows
MFk
r2- m
r
Ek
-
rk
Set Vk(Y)
(Xk' X~) <
1B~-2(Xk) X B~k (xl.)
and
--t
223
00
~
(8t Uk)2dV:S I,
as k --t
00.
= Uk((Xk, x~) + rkY, t) . It is obvious that
r
1B,{,-2(0) X B?(0)
I V vkl2(X, x', t)dxdx' =
and sup
(~)2-m
O
rk
E0
8·2 2m
(7)
m-2
r
) B':'L-2(0) X Bh.. (0) Tk
L I Vi vkl2dV < .!..k
(8)
i=l
Tk
By Lemma 2.3, for any R > 0, we have
R 2- m
r
1B~-2(0) x Bk(0)
IV
vkl2dV :S CA + CJRrk.
By the diagonal subsequence argument, we can get a subsequence of Vk which we also denote by Vk such that Vk --t v weakly in H 1,2(Rm, N). We have, for any
So v is a weak harmonic map. It follows that R2-
m
r I V vI dV :S CA. 1B~-2(0) x Bk(0) 2
(9)
Jiayu Li
224
By (8), we get
for any R > 0, hence,
ViV -
t 12
i=m-l
0 (i = 1,· .. , m - 2). By (9), one gets
I Vi vl 2 dV:::;
CA,
BR(O)
where C does not depend on R. Therefore, we have
By the removable singularity theorem in [16J (Theorem 3.6), we know that v may be extended to a smooth harmonic map from 52 to N. We set Wk(Y, s) = Uk((Xk,O) + rkY, t + r~s), it is clear that Wk(Y, 0) = Vk(Y). By (5) and (7), using the strong constance lemma as Lin did in [10], we can show that P(Wk' (0,0),2) < to, so we have Vk -+ v in Hl~;(B~-2(0) x Bi(O)). It follows from (7) that v t= Constant. So, Uk -+ U in Hl~;(Rm x R+, N), and we can see that U is a stationary weak heat flow. This completes the proof of the theorem. To prove the existence of the weak heat flow, Chen-Struwe [4J considered the heat flow for the penalized functional which is defined as follows. Suppose that N is isometrically embedded into RK for some K, assume that U is a tubular neighborhood of N of width 2<5 such that any point p E U has a unique nearest neighbor q = 7rN(P) E N, dist(p, q) = dist(p, N) and such that the projection 7rN : U -+ N is smooth. Let X be a smooth, nondecreasing function such that X(s) = s for s :::; <5 2 and X(s) = 2<5 2 for s ;:::: 2<52 . The penalized functional is defined by
Ek(u) = E(u)
+k
r X(d (u, N))dV JRm 2
The existence and convergence of heat flows
225
where d(u, N) = dist(u, N). It is clear (cf. [4]) that for any initial data Uo E Hl,2(Rm, N), any k = 1,2" ", there exists a solution Uk to the heat equation 2
£:l UtUk - D.uk
d ( d (Uk' N) ) = 0 + kX '( d2( Uk, N )) -d Uk
2
(10)
with Uk(', 0) = uo(')' The following theorem can be proved by an argument similar to the one used in the proof of Theorem 2.6 (d. [13], Section 7) . Theorem 2.11 Let Uk be a sequence of solutions of the equation (10) with initial data Uo satisfying E(uo) < 00 . If the target manifold N does not carry any harmonic spheres 52, there is a subsequence of Uk which we also denote by Uk such that Uk -t U in HI~~(Rm x R+, N). And U is a stationary weak heat flow.
3
Convergence
In this section, we prove the existence of stationary harmonic maps in the case that the target manifold N does not carry any harmonic spheres 52. Theorem 3.1 Suppose that u(x, t) is a stationary weak heat flow with E(u(·, 0)) < 00. If the target manifold N does not carry any harmonic spheres 52, there is a sequence tk such that uC tk) -t Uoo in HI~~(Rm x R+, N). And Uoo is a stationary harmonic map. Proof: Since u(x, t) is stationary, we have 2
1 au i to
o
(~)2dV dt
Rm
ut
So there is a sequence tk -t
+ E(uC to)) 00
such that
= E(u(·, 0)) .
(11)
Jiayu Li
226
We may assume that
u(x , tk) -+ uoo(x) weakly in H 1 ,2(Rm, N) . To show our theorem , it suffices to prove that
u(x, tk) -+ uoo(x) in H:~~(Rm, N). We set
By Theorem 2.4, we know that
u(x, tk) -+ uoo(x) in Hl~~(Rm \ E, N). And it is not difficult to check that Hm-2(E) < 00. We assume that 1\7 uI 2 (., tk)dx -+ 1\7 u oo I 2 (·)dx + v in the sense of measure as k -+ 00, where v is a nonnegative Radon measure in Rm supported in E. We need only show that v(E) = O. Lemma 3.2 The blow up set E is a Hm-2- rectijiable set.
Proof: We choose X(y)
= ((r)r %T' where r = Iy -
xl and
if r ::; S', (( r) = (s - r) / (s - S') if S' < r < s, { o if r 2: s. I
Inserting X in (1), letting
S'
-+ s, we have
1\7 u(y, tk)1 2 dy - s r i \7 u(y, tkWda +
(m - 2) r J B.(x)
+ 2s r J8Bs(X)
Iau(y, tk) 12 dy = 2 ar
J 8B.(x)
l
(r au(y, t k)) au(y, tk) dy, J Bs(x) ar at
so,
dE(u , (x , tk) , r) dr > 2r 2 - m r Iau 12 da _ 2r 1 - m r (r au(y , t k)) au(y, tk) dy . J 8B r ar J Br{x) ar at
The existence and convergence of heat flows
227
By Holder inequality, we get
dE(u, (x, t k), r) dr 1 > 2r 2ml18U2 -8 I dcr-2(E(u,(x,t k),r))2(I(u, (x,tk),r))2.1
r
{)B r
Integrating with respect to r from k --+ 00, we get
r~-m
( } B r2 {x)
~ r~-m
rl
to r2 (0 <
rl
< r2) and letting
1\7 uoo l2dV + r~-ml/(Br2(X)) 1\7 uoo l2dV + r~-ml/(Brl (x)).
{ } Brl (x)
Since lim r 2- m (
r~O
= 0 for H m- 2 - a.e. x E Rm,
1\7 uoo l2dV
JBr{X)
we can see that, for Hm-2_a.e. x E ~, limr~o r 2- m fBr(x)
1/
exists.
Choosing ¢ E Co(Br(x)) with ¢ = 1 in B~(x), I 'V ¢I :::; ~ in Definition 2.1, we obtain, for any tk - r2 < s < tk + r2,
Ir 2- m {
¢I 'V uI 2(., tk)dV - r 2- m {
} Br{x)
¢I 'V u1 2(-, s)dVI
} B r {x)
k
l~uI2(. , t)dVdtl
< r 2_m1 jt ( JBr{X)
s
+ crl-llnljtk { s
t
l'Vull~uldVdtl t
} Br(x)
< r2_mltk+r2 {
l~uI2(. , t)dVdt
tk- r2 } Br(X)
t
+ crl-m(ltdr2 (
tk-r2 } Br(x)
tdr2
( I~u 12dV dt) t. tk-r2 } Br{x) t
I 'V ul 2dV dt)! (l
So we have lim E( u, (x, tk), r)
k~oo
~
4·
;
r
m
-2
lim inf P( u, (x, tk), -2 ). k~oo
(12)
Jiayu Li
228
C C '" l'lmr-+o r 2-mJ'Br{x) 1/ > ~ Th erelore, lor Hm-2 -a.e. x E LJ, _ 4.2m ':' 2 ' Applying Theorem 5.6 in [14], we show that ~ is a Hm-2- rectifiable set. This proves the lemma. We assume that 1/ = O(x)Hm-2l~, where O(x) is upper semicontinuous in ~ with c ::; O(x) ::; C for Hm-2_a.e . x E ~, c is a positive constant depending only on m and EO , C is a positive constant depending only on m and E(u(· , O)). So, O(x) is Hm-2 approximate continuous for Hm-2_a.e . x E ~. The following Geometric Lemma can be proved by an argument similar to the one used in the proof of Lemma 2.4 in [1OJ.
Lemma 3.3 Suppose that x E ~ and O(x) is Hm-2 approximate continuous at x. Then there exists rx > 0, such that, for any 0 < r < r X1 we can find m - 2 points Xl,"', Xm-2 in Br(x) n ~ so that, (1) JO(Xj) - O(x) < Er , for j = 1, ... ,m- 2, where Er ---t 0 as r ---t 0; and (2) JXIJ2:: Smr , dist(xk,x+ Vk- 1 ) 2:: Smr , fork = 1, .. ·,m-2, where Vk- l is the linear space spanned by {Xl - X, ... , Xk-l - X}. Here Sm > 0 depends only on m . J
Using the Geometric lemma, as we do in the proof of Lemma 5.2 in [13J we can prove the following lemma.
Lemma 3.4 Let T E
T~.
We have
Set
Fkf(X) = for
X
t JB;{O)
J
\7T uJ2(X, x', tk)dx'
E ~. We consider the Hardy-Littlewood maximal function
MFk E(x)
of Fh(X), which is defined by MFk E(x)
= sup r 2- m ( Fkf(x)dx . O
The existence and convergence of heat flows
229
By the weak type (1, 1) inequality for M FkE (x), we have
for any ..\ > O. By Lemma 3.4, we have
thus for any l > 0,
Hm-2(U~=1 n~=no Uko=l n~ko {x
E EIMFk(;t) (x)
2':
~}) = o.
By the partial regularity result (cf. [6], [9], [12]), we can find Xl E E \ 512 (tk) c B~-2(0), such that for any no > 0 and any ko > 0 there are nl > no and kl > ko satisfying
(13) and u(x, t k) is smooth near (Xl, x f) for all Xf E Br(O), and (Xl, x f ) fI51 (tk). 2 We claim that, for all k sufficiently large, we may find 8k -t 0 such that max
X'EB~(O)
8~-m
r
} B";',.-2(Xk) XB
'ik (x')
I \l uI 2(x, Xf, tk)dxdx f =
8 .
E~2
.
(14)
m
In fact, since u(x, tk) is smooth at (Xk' Xf), for any given k and for 8 < 8(k), we have
On the other hand, by (12) we have EO
lim E(u, (x, tk), r) 2': 4 . 2m -2·
k-+oo
230
Jiayu Li
So, for fixed
~
> 0 and sufficiently large k,
Therefore we can choose ~k > 0 so that (13) holds. By (13) and (14), since (Xk' x') tt. Sl2 (t k), we can find Ek --t 0, Tk --t 0, (Xk, X~) E E and a subsequence of Uk , which we also denote by Uk for simplicity, such that
MFk
T~-m
(Xk,
r
JB~-2(Xk) X B¢k (x;')
x~) < ~ (Ot U)2(., tk)dV ::; 1,
and Ek --t 00
Tk
Set Vk(Y) = U((Xk, x~)
+ TkY , tk).
as k
--t 00 .
It is obvious that
JrB,{,-2(0) x Br(0) I 'V vkI2(X, x', t)dxdx' = and sup ( -T)2 - m O< r < l
Tk
1
B";:-2(0) X Bh. (0) rk
/
8 .
E~2m
m-2 ~ 'Vi Vk /2 dV < -. 1 L..J i=l k
(15)
(16)
rk
By Lemma 2.3, for any R > 0, we have
By the diagonal subsequence argument, we can get a subsequence of Vk which we also denote by Vk such that Vk --t v weakly in
The existence and convergence of heat flows
231
So v is a weak harmonic map. It follows that
By (16), we get
for any R > 0, hence, gets
\!iV
= 0 (i = 1"
.. ,m - 2) . By (17), one
where C does not depend on R. Therefore, we have
By the removable singularity theorem in [16] (Theorem 3.6), we know that v may be extended to a smooth harmonic map from 8 2 to N. We set Wk(Y, s) = U((Xk'O) + TkY, tk + T~S), it is clear that Wk(Y,O) = Vk(Y)' By (12) and (14), using the strong constance lemma as Lin did in [10], we can show that P(Wk' (0,0),2) < EO,
232
Jiayu Li
so we have Vk ---* v in HI~;(B~-2(0) x Bi(O)). It follows from (15) that v ¢. Constant. This completes the proof of the theorem. By Theorem 2.11, Theorem 3.1 and the regularity theorem for stationary harmonic maps (Theorem A in [10]), we obtain the following theorem. Theorem 3.5 Suppose that the target manifold N does not carry any harmonic spheres 52. For any Uo E Hl,2 (Rm, N), there is a stationary weak heat flow with initial data Uo . And the weak heat flow subconverges to a stationary harmonic map U oo . If in addition, the target manifold does not carry any harmonic spheres 51 (l = 3, ... ,m - 1) U oo is a smooth map.
Acknowledgments This work is supported by NSF grant.
References [1] Bethuel, F., On the singular set of stationary harmonic maps, Manu. Math., 78(1993), 417-443; [2] Chang, K., Ding, W. and Ye, R., Finite Time Blow-up of the Heat Flow of Harmonic Maps, J. Diff. Geom. 36(1992), 507515; [3] Chen, Y. and Ding, W.-Y., Blow up and global existence for heat flows of harmonic maps, Invent. Math. 99(1990), 567578;
[4] Chen, Y. and Struwe, M., Existence and partial regularity for heat flow for harmonic maps, Math. Z. 201(1989) 83-103;
[5] Coron, J.-M. and Ghidaglia, J.-M ., Explosion en temps fini pour Ie flot des applications harmonique, C. R. Acad. Sci. Paris 308(1989), Serie I, 339-344:
The existence and convergence of heat flows
233
[6] Chen, Y., Li,
J. and Lin, F. H., Partial regularity for weak heat flows into spheres, Comm. Pure Appl. Math. Vol. XLVIII(1995), 429-448;
[7] Ding W.-Y., Blow up of solutions of heat flow for harmonic maps, Adv.in Math. 19(1990), 80-92;
[8] Eells,
J. and Sampson, J. H., Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86(1964), 109-160;
[9] Feldman, M., Partial regularity for harmonic maps of evolution into spheres, Comm. Partial Diff. Equat. 19(1994), 761-790;
[10] Lin, F. H., Gradient estimates and blow up analysis for stationary harmonic maps I, Preprint;
[11] Lin, F. H. and Wang, C., Harmonic and quasi-harmonic spheres, Preprint;
[12] Liu, X., Partial regularity for weak heat flows into general compact Riemannian manifolds, Preprint;
[13] Li,
J. and Tian, G., The blow up locus of heat flows for harmonic maps, Preprint;
[14] Priess, D., Geometry of measures in Rn; Distribution, rectifiability, and densities; Ann. of Math., 125(1987), 537-643;
[15] Struwe, M., On the evolution of harmonic maps in higher dimensions, J. Diff. Geom. 28(1988), 485-502.
[16] Sacks,
J. and Uhlenbeck, K., The existence of minimal immersions of two spheres. Ann. Math. 113(1981), 1-24;
Some aspects of semilinear elliptic boundary value problem* Shujie Li+
Institute of Mathematics, Academia Sinica, Beijing 100080, P. R. China Abstract. In this paper, we will summary some new advances in existence theory for solutions of the following semilinear elliptic boundary value problem - b. u = g(x, 8) { u= 0
in n on an,
where n c ]RN is a bounded domain with a smooth boundary. We will discuss the superlinear, asymptotically linear and jumping nonlinear problems according to growth offunction g(x, u) with respect to u. By using Morse theory (we stress the computation of critical groups at degenerate critical point in particular), minimax theory (including linking and local linking arguments), sub- and super-solutions method and the invariance property of order intervals under the descent flow, and Mountain Pass Theorem in Order Intervals, we will state a series of new results on existence of multiple solutions and sign changing solutions for the elliptic boundary value problems. For strong resonance problems, we have introduced a new approach, Morse theory approach, to establish the multiplicity results. We will introduce this in other paper, see [Lil, HLWl, LiWa2J .
1. Super linear Problem Throughout this paper we consider the following semilinear elliptic boundary value problem - b. u = g(x, u), in n (1.1) { u = 0, on an where n c ]RN is a bounded domain with a smooth boundary. We make the following hypotheses on g. (gd 9 E C(IT X ]Rl,]Rl), there exist Q, c > 0 such that
Ig(x, t)1 where 1 <
Q
< ~ ~ ~ as
:s c(l + IW')
n ;::: 3.
* This work was carried out while author was visiting the Morningside Center of Mathematics, Chinese Academy of Sciences. + Supported in part by National Science Foundation of China and Foundation of Chinese Academy of Sciences.
234
Some aspects of semilinear elliptic boundary value problem
(g2)
235
There exists go E R such that . g(x, t) go = 11m - Itl-+o
£
unhormly in XED,
t
go < AI. (g3)
There exist ()
> 2, M > 0 such that
o < (}G(x, t)
~
t . g(x, t)
for
It I ~
M.
(g3) is a famous superlinear condition given by P. Rabinowitz. The first important result for (1.1) is the following. Theorem 1.1 Let (gt) - (g3) hold. The (1.1) has at least one nontrivial solution. This theorem was given by A. Ambrosetti and P. Rabinowitz (see [AmR1]). In [Cha2], K. C. Chang weakened the conditions (g2), (g3) by -1' g(i' t) < Imt-+O _
\
t
uniformly in xED,
+t
uniformly in xED,
Al -
· -+ g(i' t) > \ 1Im _ Al t o
and
(}G(x, t)
~
tg(x, t),
"Ix E 0 as
It I ~
M.
More complicated problem is the following - 6. u { u = 0,
+ a(x)u = g(x, u),
in 0 on aD
(1.2)
where a(x) E LOO(O). In [Ra2], P. Rabinowitz proved the following theorem by using the Generalized Mountain Pass theorem . If (gl), (g3) hold and a(x) = A, (g2)' g(x, t) = o(ltl) as t --+ O. (g4) tg(x, t) ~ 0 "Ix E 0, t E IR. Then for each A E R, (1.2) possesses a nontrivial solution. Using local linking argument Theorem 1.2 was improved by J. Q. Liu and S. J. Li (see [LiuLil]). The following more general result was given by S. J. Li and M. Willam. Theorem 1.2
Shujie Li
236
Theorem 1.3 [LiWil] . Under the assumptions (gd, (g2)', (g3), if 0 is an eigenvalue of - 6, +a (with Dirichlet boundary condition) , assume also, for some 0 > 0, either
G(x, u)
:=
l"
g(x, s)ds
~ 0 for lui:::; 0,
x
E
n,
or
G(x,u):::; 0 for
lui:::; o, x E n.
Then (1.2) has at least one nontrivila solution. Next we consider the following problem {
- 6, u + a(x)u u=O
= )..g(x, u)
in n on
an.
(1.3),\
H. Brezis and 1. Nirenberg proved the following
Theorem 1.4 (g5)
[BrN1] . Suppose that 9 satisfies (gl), (g2)' and
g(x , t) 0 · 11m - - < . Itl--+oo t If 0 is an eigenvalue of - 6, +a, we assume also that, for some 0 > 0
G(u)
~
0 for
lui:::; o.
Then for every ).. sufficiently large, there are at least two nontrivial solutions of (1.3);,. S. J . Li and M. Willem considered the case G(u) :::; O. Theorem 1.5 [LiWi1] . Under the assumptions of Theorem 1.4. If 0 is an eigenvalue of - 6, +a, we assume also that, for some 0 > 0
G(u) :::; 0 for
lui:::; o.
Then for every ).. sufficiently large, there are at least two nontrivial solutions of (1.3};, . Let X be a Banach space with a direct sum decomposition X = Xl Ef:) X2. The function IE C(X, JR.) has a local linking at 0 with (Xl, X2) if, there exist 0,00 > 0 such that (h) I(u) ~ 0, u E Xl, lIull:::; 0, (12) I(u) ~ 00> 0, u E Xl, lIull = 0, (13) I(u):::; 0, u E X 2, Ilull:::; o.
Some aspects of semilinear elliptic boundary value problem
237
This condition was first introduced and used to find critical points by J . Q. Liu and S. J. Li for the case dimX 2 < 00 (see [LiuLil]). More weak assumption were introduced by E. A. Silva, and used by H. Brezis and L. Nirenberg to prove the existence of a nontrivial critical point, in the case (11), (13) but dim X 2 < 00 (see [Sill and [BrNl]). In [LiWil], S. J . Li and M. Willem droped dimX 2 < 00 and dealt with the case, I is strongly indefinite functional. We shall use the following compactness conditions for the strongly indefinite functionals. Consider two sequences of subspaces
Such that Xj
= U xnj,
J. = 1, 2 .
nEN
For every multi-index a = (aI, (2) E N2 , we denote by XOt the space X~l EBX~2 ' Let us recall that a :::; /3 {:=:::} a1 :::; /31, a2 :::; /32 ' A sequence (an) C N2 is admissible if, for every a E N 2 there is mEN such that n ~ m ===> On ~ a. We denote by lOt the function I restricted to X Ot . Definition. Let c E IR and I E C 1 (X,IR). We say that the function [ satisfies (PS)~ condition if every sequence (UOtJ such that (an) is admissible and UOtn E X Otn , I( u Otn ) -+ c, I~J uOtJ -+ 0 contains a subsequence which converges to a critical point of I. Definition. Let IE C 1 (X, IR). We say that I satisfies the (PS)" condition if every sequence (UOt n ) S.t. (an) is admissible and UOtn E X Otn , sUPnl(uOtJ < 00, I~n (UOt n ) -+ 0, contains a subsequence which converges to a critical point of I. Remark. (PS)" condition was first introduced in [BaBel] . The following theorem we choose from [LiWil]. Theorem 1.6 Suppose IE C 1 (X, IR) satisfies the following assumptions: (Ad I has a local linking at 0 with (Xl, X2) (without (12)), (A 2 ) [satisfies (P S)" condition, (A3) [maps bounded sets into bounded sets, (A4) [is bounded below and d:= infx f < O. Then I has at least two nontrivial critical points. Theorem 1.6 is a more general case of the following theorem given by K. C. Chang.
Shujie Li
238
Theorem 1.7 Let X be a Hilbert space and I E C 2 (X,JR). I satisfies (PS) condition and bounded from below. 0 is not a minmizer of I. If 0 is a nondegenerate critical point with finite Morse index. Then I has at least two nontrivial critical points. Remark. In Theorem 1.6 0 can be degenerate and the Morse index at 0 can be infinity. Now we consider the following problem
{
- 6. u
u=o
= g(x, u) -
h
in 0 on ao
(1.4)
Theorem 1.8 [Cha3]. Suppose that (gl), (g3) hold, and 2 (hr) hE Ln -f'2 (0) is nonnegative, h -:t. o. g(x, t) ~ 0 V(x, t) E 0 X JR1, g(x,O) = 0 and there exists to > 0 such that g(x, to) > o. Then (1.4) has at least two nontrivial solutions. The idea to prove Theorem 1.8 is by using the sub-and super-solution method and Mountain Pass Theorem. Now we consider the multiple solutions of (1.1) . When 9 E C 1 (IT x IRl), it is well known that (1.1) has a positive solution and a negative solution (see [AmRl]) . The following theorem given by Z. Q. Wang point out that (1.1) possesses the third solution. Theorem 1.9 [Wanl]. Suppose that (gr), (g3) hold, and 9 E C 1 (IT X JR1), g(x,O) = gt(x, 0) = o. Then (1.1) has at least three nontrivial solutions. Z. Q. Wang proved this theorem by two methods, one is Morse theory, the other is linking and minimax argument. 1 9 E C (IT x IR.1) can be droped. See recent papers given by J. Q. Liu [Liu2], and S. J. Li -Z. Q. Wang [LiWal]. If we do not require g(x,O) = O. It is possible to discuss the existence of four solutions. The following theorem given by Z. L. Liu and J . X. Sun. Theorem 1.10 Suppose that 9 satisfies (gr), (g3), and (1.1) has a pair of sub-and super-solutions. If there exists m > 0 such that g(x, t) +mt is increase in t. Then (1.1) possesses at least four solutions. Now we state some results for existence of sign--changing solutions. For superlinear case the following theorem was given by T. Bartsch-Z. Q . Wang [BaWal]' and A. Castro-J . Cossio-J. Neuberger [CCN] independently.
Some aspects of semilinear elliptic boundary value problem
239
Theorem 1.11 Suppose that 9 satisfies (g2) , (g3) and (gd' There exist c, 0: > 0 such that 0:
n+2 n-
< --2'
1
9 E C (fl
x JR, JR) .
(g3)' There exists m E JR such that g' (x, t)
> m 'Vt
E JR,
x E fl.
Then (1.1) has at least three nontrivial solutions: one is positive, one is negative and the third is sign-changing. In [LiWa1J, S. J . Li and Z. Q. Wang droped (g3)' and substitute more weak condition (g1) for (gd· Theorem 1.12 [LiWa1) . Suppose that 9 satisfies (gd, (g2), (g3) . Then (1.1) has at least three nontrivial solutions: one is positive, one is negative and the third is sign-changing. In [LiWa1J, a Mountain Pass Theorem in Order Intervals was given in which the position of the mountain pass point is precisely given in term of the order structure. Remark. In [Liu2) J. Q. Liu also gave a proof of Theorem 1.11 without assumption 9 E C 1 . Another simple proof of Theorem 1.11 was given in [LiZha1). Open problem: Does (1.1) have more solutions under the assumptions of Theorem 1.12? Theorem 1.13 Under the assumptions (gd , (g3), if there exist tl 0, t2 > 0 such that g(x , tl) ~ 0, g(x, t2) ~ 0, 'Vx E fl, and 9 satisfies,
(92)'
A2 < go := lim 9(X , t) Itl--tO t
<
~ a( -6) ,
(96) there exists m > 0 such that g(x , t) + mt is increase in t. Then (1.1) has at least seven nontrivial solutions, including two positive solutions , two negative solutions and three sign-changing solutions. This theorem was proved by Dancer and Y. H. Du[DaDu1) using Conley index, by K . C. Chang [Cha4j using Morse theory, and S. J . Li and Z. T . Zhang [LiZha1) using invariance property of flow . Using Mountain Pass Theorem in Order Intervals we can give a very simple proof of Theorem 1.13 under more weak assumptions. Let E be a Hilbert space and PE C E be a closed convex cone. Let X C E be a Banach space which is densely embedded to E . Let P = X n PE and
Shujie Li
240 o
assume that P has nonempty interior p,p 0. We assume any order interval is finitely bounded. eI> is a functional from E to IR and satisfies the following assumptions: (eI>d eI> E CI(E, IR) and satisfies the (PS) condition in E and deformation property in X. eI> only has finitely many isolated critical points. (eI>2) The gradient of eI> is of the form VeI> = Id - KE where KE : E -t E is compact, KE(X) c X, K = KElx -t X is continuous and strongly order preserving, i.e., U > v ===} K(u) » K(v) for all u,v E X, where u » v {:=} o U -
v EP.
(eI>3)
eI> is bounded from below on any order interval in X.
Proposition. [LiWa1). Suppose eI> satisfies (eI>d, (eI>2)' (eI>3), and!! < u is a pair of sub-solution and super-solution of VeI> = 0 in X. Then there exists a negative pseudo gradient flow TJ(t, .) in X such that [!!, u) is positively invariant under the flow and TJ(t,·) points inward on 8[!!,u)\{y,u}. The following is the Mountain Pass Theorem in Order Intervals. Theorem 1.14 [LiWa1) . Suppose eI> satisfies (eI>d, (eI>2)' (eI>3). Suppose there exist four points in X, Vl < V2 , Wl < W2, Vl < W2, [VI , V2) n [WI, W2) = 0 with Vl :S K Vl, V2 > K V2, Wl < K WI, W2 2: K W2. Then eI> has a mountain pass point Uo E [Vl, W2)\ ([Vl, V2) U [Wl, W2)). More precisely, let Vo be the maximal minimizer of eI> in [VI, V2) and Wo be the minimal minimizer of eI> in [Wl, W2), if Vo < Wo then Vo «uo «wo. Moreover Cl(eI>,uo) , the critical group of eI> at Uo is nontrivial. Remark. In Theorem 1.14 one can suppose that VI, W2 are solutions of VeI> = O. Using the Proposition and Theorem 1.14 we have the following Theorem 1.15 [LiWa1) . In Theorem 1.13 if one substitute (gl) for (gd. Then (1.1) has at least two positive solutions ut < ut, at least two negative solutions u > ui, and at least three sign- changing solutions Us, U6, U7 with u < Us < ut , u < U6 < ut where ut , u are the minmizer of the action integral of (1.1), and Us is the mountain pass point. In Theorem 1.13 the condition (96) can be droped. Let
a
a
a
a
E = Ho(o') , X = CJ(O,). (96) guarantees the order preserving property of the negative gradient flow for eI>. Since VeI> = Id - K where K is a compact
241
Some aspects of semilinear elliptic boundary value problem
operator and is strong order preserving. If (g6) does not hold, K may not satisfy the order preserving property. In order to overcome this difficulty we can construct a special vector field which may not be a pseudo gradient vector field. Nevertheless, we can show the flow under this vector field still satisfies the deformation property and prossesses some prescribed invariance property. Let us assume (~2)' ~ E C 2(E, JR) and the gradient of ~ is of the form \7~(u) = AouKE(U) where KE : E ~ E is compact, KE(X) C X and the restriction KE on X, K : X ~ X is of class C 1 and is strongly order preserving, i.e., u > v ==>
»
o
»
v
if
IIAil1(U)II:::; 1
~ E is locally Lipschitz operator such that Ail 1 : E ~ E exists, Ail 1(X) C X and the restriction Ail 1IX : X ~ X is of class C 1 and strongly order preserving. Furthermore, there is M > 0 such that Ao(u) is linear for all u E X and Ilullx :::; M. Finally, we assume that there exists ao > 0 such that
K(u)
K(v) for all u, v E X, where u
IIAil1(u)11
~
aollull
u - v EP; Ao : E
¢:::::}
and
(Ail 1 (u),u) ~ II A il 1 (u)112. In [LiWa1] it was shown that under the assumption (~2)/ one can construct a flow TJ(t, u) along which ~ decreases and satisfies a deformation property, and under which ±P and ±F are both positively invariant, and finally TJ(t, u) leaves any order interval of the form [~, u] invariant provided :!!, u is a pair of sub-and super-solutions of \7~(u) = 0 with 11:!!llx :::; M and Ilullx :::; M. For problem (1.1), let for lui:::; Mo for lui ~ Mo + 1 and m( u) is C 1 and monotonically increasing. Here mo, Mo and p are constants, 0: < p < 2* = ~ ~, 0: was given in (g1). If mo and Mo are large g(x,u) := g(x, u) + m(u) is strictly increasing. Now consider
%
{ ~(u)
-.6. u + m(u)
= g(x,u)
u=o
n
in on 8n
(1.5)
can rewrite
~(u) = .!.
r 1\7 ul dx + inr M(u)dx - inr G(x, u)dx, 2
2 in
where M(u)
= iou m(s)ds,
G(x, u)
= iou g(x, s)ds.
\711>(u) = Ao(u) - K(u)
Then
u E E,
\7~ has a form
Shujie Li
242
where K(u)
= (-6)-l g(X,U) (Ao(u), v)
is strongly order preserving, and
= lt17u \l v + m(u)v)dx
i.e., Ao(u) = u + (-6)-l m(U) . It is easy to check that 4)(u) satisfies (4)2)'. Using the Mountain Pass Theorem in Order Intervals one can prove Theorem 1.16 [LiWaI]. Without condition (g6) Theorem 1.13 still holds. Next we deal with the Dirichlet problem with a changing-sign nonlinearity. Consider in 0 - 6 u = AU + h(x)g(u) (1.6h. { u=o on ao where 0 c ]RN is a bounded open set with smooth boundary, h E G"'(fl) changes sign in 0, a > 0, 9 is a Gl function and satisfies (g2)' and following condition, (g7) For some a, 2 < a < 2*
Suppose that h has a "thick" zero-set (hr) {xlh(x) > o} n {xlh(x)
< o}
=
0.
As in [OuI] one define A*
= inf{11 \l vll~1 Ilvlb = 1,
in
h(x)lvIPdx
= o}.
We introduce the usual action functional
where G(u)
= iou g(s)ds.
We will seek nontrivial critical point of l>,(u) on
HJ(fl) which is equivalent to find weak solution of (1.6}>.. Let Aj(O) be the jth eigenvalue of -6 with Dirichlet boundary data. Let 'Pj be the eigenfunctions corresponding to the eigenvalue Aj. Then Theorem 1.17 [LiWaI]. Assume (g2)', (g7), (hr) hold, and h E G"'(IT) changing sign in o. (1) If A < Al (fl) then (1.6)>. has at least three nontrivial solutions, one is positive, one is negative, and the third is sign-changing.
Some aspects of semilinear elliptic boundary value problem
243
(2) If in addition lim u-+o
for some q
g(u)
lul Q- 2 u
=a>O
(1.7)
> 2, and (1.8)
then there exists X> 0, X~ A. such that if A2(O) < A < X, A ~ a( -.6.), (1.6)A has at least seven nontrivial solutions. More precisely, (2a) (1.6h has at least two positive solutions ut and ut where h(ut) < 0, ut is a local minimizer of JA(u) . (2b) (1.6h has at least two negative solutions u3' and ui where JA(ui) < 0, ui is a local minimizer of JA(u) . (2c) (1.6h has at least three sign-changing solutions U5, U6, U7 where U6 is a mountain pass point of JA(u), J A(U6) < 0, ui < U6 < ut, ui < U7 < ut and U7 is outside [ui, ut] . (3) If in addition (1.7) (1.8) hold, AdO) < A < X, A ~ a( -.6.), then (1.6h has at least five nontrivial solutions. More precisely, (2a) and (2b) hold and there is a sign-changing solution U5. ReInark. In [AIDI] . Alama and Del Pino proved part (1) of Theorem 1.17 except the assertion on the sign-changing solution, and proved part (3) of Theorem 1.17 also except the assertion on the sign-changing solution but for the case Al(O) < A < X < A2(O) . In [BeCDN], Berestycki, Capuzzo Dolcetta and Nirenberg have studied the existence of positive solutions for equations more general than (1.6h.
2. Asymptotically Linear Problems We still consider (1.1) . If there exists goo E IR such that lim Itl-+oo
g(x , t) - goot ·t
=0
'r I ' unllorm y In
X
r.
EH
.
we called g(x, t) is asymptotically linear. Let
f(x, t)
:=
g(x, t) - goot.
Suppose that there exists go E IR such that 9 -
0-
. g(x, t) - g{x, 0) I1m Itl-+o t
uniformly in x E O.
Shujie Li
244
Let
Bo Boo
:= := -
!:::.. u + gou, !:::.. u + gooU'
By mo, moo we denote the dimension of negative subspace of Bo and Boo respectively. By mo, m~ we denote the dimension of nonpositive subspace of Bo and Boo respectively. In [AmaZ1] Amann and Zehnder proved the following Theorem 2.1 Suppose that 9 E C I (IT X RI) , g(x ,O) = 0, go , goo exist, goo (j. 0"( -!:::..) . If moo (j. [mo, mol then (1.1) has at least one nontrivial solution. This theorem was proved by using Conley Index. Later in [Chal], K. C. Chang gave a simple proof by Morse Theory. In [LaSo1] Lazer and Solimini considered more general case. By using the local linking argument , in [LiLiu1J, J. Q. Liu and S. J. Li dealt with the resonant case at zero. Let X = HJ(O)
J(u) = If go
= Ai
~ In I 'V ul 2dx -
In G(u)dx
(2.1)
then we can rewrite (1.1) as following
- !:::.. u + gou = gou + g(x, u) { u=O
m 0 on .80
(2.2)
We make the following assumption (gs)± There exists 6 > 0 such that 1
±(2got2 -
10t
g(x, s)ds) > 0 as
It I ~ 6.
Let X
2
Xl
= {Span{'PI ""
,'Pj-d
Span{'PI, '" ,'Pj}
= Xt-
as (gs)+ holds , as (gs) _ holds,
(in HJ(O)).
One can check that if (gs)± holds, then there exist r > 0,60 > 0 such that
J(u) > 0 as u E Xl, J'(u) 2: 60 > 0 as u E Xl, { J(u) ~ 0 as u E X 2 ,
Ilull ~ r, Ilull = r, lIull ~ r.
(2.3)
Theorem 2.2 [LiLiu1] Under one of the following conditions (1.1) has at least one nontrivial solution.
Some aspects of semilinear elliptic boundary value problem
245
(1) 90 , 900 (j. a( -~), there exists at least one eigenvalue between 900 and 90 ·
(2) 900 (j. a(-~), 90 = >'j E a(-~), (98)± holds and dimX2 i:- moo . Theorem 2.1 and 2.2 are different. But in Theorem 2.2 it is possible to discuss the case moo E [mo, mol provided that (98)± holds. In [LaSol] the resonant case at infinity was discussed Theorem 2.3 Let X be a Hilbert space, X C 2 (X, JR) and satisfies (PS) condition. If
= YEBZ , dim Y
< 00, I(u)
E
inf I(u) = d > -00,
uEZ
I(u) ~
-00
as
lIull ~ 00
u E Y.
Suppose () is a isolated critical point of I. D2I(()) is the Fredholm operator. Either dim Y > mo or dim Y < mo . Then I has a nontrivial critical point u* . In [BaLil] T . Bartsch and S. J. Li got more information. Theorem 2.4 Then
Under the assumptions of Theorem 2.3, let dim Y
= k.
where C k (1,00) is the k-th critical group of I at infinity, defined by Ck(I, 00) := Hk(X , Ia) , where Ia = {x E XII(x) ~ a}, a < infuEK I(u), K is the critical point set of I , and Hk(X, Ia) is the singular k-relative homology group. In [LiWil] S. J. Li and M. Willem substituted following (98)± for (98)± in Theorem 2.2, (98)± There exists 8
> 0 such that ±(i90t2
-lot
9(X , s)ds) 2: 0 for
It I ~ 8.
Concerning the multiple solutions, in [ChaLL1], K. C. Chang, S. J. Li and J . Q. Liu Proved the following
=
Theorem 2.5 Suppose that 9 E Cl(IT X JRl), 9(X,O) 0 , 90 < >'1,900 > >'2' Then under the one of the following conditions (1.1) has at least three
nontrivial solutions (including one positive, one negative) . (1) 900 (j. a( -~), (2) 900 E a( -~), ¢(x, t) := 9(X, t) - 900t is bounded and satisfies the Landesman-Lazer condition
Shujie Li
246
where (x, t)
= lot ¢(x, s)ds,
Span{CP1, " ', CPm}
= Ker( -
6. -goe'!).
Theorem 2.6 Suppose that 9 E C 1(n x JR1), g(x,O) = 0, Al < go < Ak < goo, goo satisfies assumption (1) or (2) in Theorem 2.5. If there exists to # such that g(x, to) = \Ix E Then (1.1) has at least three nontrivial solutions. If A2 < go < Ak < goo \Ix E Then (1.1) has at least four nontrivial solutions. Similar result also was given by Dancer and Y. H. Du (see [DaDul]) . In [BaWal] T . Bartsch and Z. Q. Wang pointed out that the third solution of Theorem 2.5 is sign--changing.
°
°
n.
n.
Theorem 2.7 Suppose that 9 E C 1(n x JR1), g(x,O) = O,go < A1,goo > A2, goo satisfies the assumption (1) or (2) in Theorem 2.5. Then (1.1) has a sign--changing solution. Remark. Using the peseudo gradient flow introduced in [LiWal) and combining the sub-and super-solution argument the assumption 9 E C 1 in Theorem 2.5, 2.6 and 2.7 can be droped. The following theorem given by T . Bartsch, K. C. Chang and Z. Q. Wang. Theorem 2.8
[BaCWl). Under the assumptions of Theorem 2.7 if 9 also
(t'
satisfies g' (x, t) 2:: 9 t) \Ix En, t E lit Then (1.1) has one more sign--changing solution with Morse index 2. S. J. Li and Z. T . Zhang have the following Theorem 2.9 [LiZha2] . If 9 E C 1(n x JR), g(x,O) = 0, go < A1,goo = Ak, k 2:: 2, g' (x, t) ::; r < Ak+1 . If g(x, t) - goot satisfies assumption (2) in Theorem 2.5. Then (1.1) has at least four nontrivial solutions: one is positive, one is negative, one is sign-changing. As we stated above if the nonlinearity crosses at least one eigenvalue (1.1) has at least one nontrivial solution. For the case in which the nonlinearity crosses the first eigenvalue, the existence of multiple solutions have been studied by many authors see [AhLPl), [AmI), [AmMl), [Danl), [Hi 1), [HoI), [Stl], [Til]. For the case in which the nonlinearity crosses the higher eigenvalues, if there are always two nontrivial solutions it is still an open problem. Under some additional condition, in [LiWi2), S. J. Li and M. Willem proved the following Theorem 2.10 Suppose that go ¢ eT( -6.), Aj < goo < Aj+1 < go, g'(x, t) E JR, j > 1. Then (1.1) has at least two nontrivial solutions.
C> Aj \Ix E n u
~
Some a.spects of semilinear elliptic boundary value problem
247
Theorem 2.11 Suppose that Aj < go < Aj+1 ::; Ak < goo < Ak+1, g'(x, t) ::; c < Ak+1, Vx E n, t E R Then (1.1) has at least two nontrivial solutions. It is similar to the superlinear problem. Using the Mountain Pass Theorem in Order Intervals we can prove
Theorem 2.12 Suppose that go > A2, go f/. CT( -6), goo ~ A2. If goo E CT( -6) also require the assumption (2) of Theorem 2.5. If there exist t 1, t2 E JR1, t1 < 0, t2 > 0 such that g(x, td ~ 0, g(x, t2) ::; 0 Vx E n. Then (1.1) has at lea.st two positive solutions ut < ut, two negative solutions u3" > ui and three sign--changing solutions us, U6, U7. Where ut, u3" are the minimizers of action functional I (u), ut, ui , Us are the mountain pass points of I (u). Remark. We do not ask any smooth assumption on g. For the case g E C 1(IT x JR) the related results see [Cha5], [DaDu1] and [LiZha1]. Corollary 2.13 In theorem 2.12 if A1 < goo < A2, then (1.1) has two positive solutions, two negative solutions and two sign-changing solutions. If A1 < go < A2 then (1.1) has two positive solutions, two negative solutions and one sign-changing solutions. If A1 < go < A2, A1 < goo < A2 then (1.1) has two positive solutions and two negative solutions. The idea to prove Theorem 2.12 and Corollary 2.13 was given in [LiWa1], [LiZha1]. Now we are in a position to consider the resonant problem. If go E CT( -6) or goo E CT( - 6.) we call (1.1) is resonant at origen (or at infinity) respectively. It is more delicate to compute the critical groups at degenerate critical points. Let I(u) = ~ I '1 ul 2 - ~ goou 2 + ~gooU2 - G(u),
In
In
In
~(t) =
(x, t) = g(x, t) - Akt,
lot (x, s)ds.
In [Cha4] K. C. Chang pointed out Theorem 2.14
In ~(uo)dx
Suppose that goo -t
= Ak,
(x, t) is a bounded function and
+00 as Uo E Ker (- 6 -Ak) and Iluoll
Then [ satisfies the (P S) condition and
-t
00.
Shujie Li
248 where r
= max {dimensions of nonpositive eigenspaces of
- 6. -
Ad
Remark. In [AhLP1) Ahmod, Lazer and Paul proved that under the assumptions of Theorem 2.14 problem (1.1) has a solution. In fact, using the Saddle Point Theorem it is easy to prove the result. Theorem 2.14 give more information: (1.1) has a solution u* and Cr(I, u*) -:j; O. In [Si1) and [Col) E. Silva and D. Costa generalized the result given in [AhLP1) and they pointed out that if
and
. hm
Itl~oo
~(x, t)
-2-
It I
= +00(-00).
Q:
Then (1.1) has a solution. Remark.
in ~(Uo)
In Theorem 2.4
dx -t +00.
Then
if one substitute
L~(uo)dx
-t -00
for
Cq(I,oo) = Oq,mooG where moo = max {
dimensions of negative eigenspace of - 6. - Ad, see [LiLiu2). In [BaLil) T. Bartsch and S. J. Li droped the boundedness of 0 and c: E (0,1) such that (I'(u), v) ~ 0 for any u = v + wE V EB W with lIull ~ Rand Ilwll ::; c:llull. (2) Cq(I,oo) =:: Oq ,moo+v oo provided that I satisfies the following condition at infinity: (AC,;,): There exist R > 0 and c: E (0,1) such that (I'(u),v) ::; 0 for any u = v + wE V EB W with lIull ~ Rand Ilwll ::; c:llull.
Some aspects of semilinear elliptic boundary value problem
249
Now we apply Theorem 2.15 to (1.1) . Let ¢(x, t) = g(x, t) - goot and ¢ satisfies the following condition. (¢~) There exist
In
In
¢(x,t)t/IW·:::; lim sup ¢(x, t)t/IW· :::;
uniformly for a.e. x E fl. Theorem 2.16 [LiZol]. Suppose that (
(2) If 2 such that Wl(X):::; liminf'¢(x,t)t/IW:::; limsup'¢(x,t)t/IW:::; W2(X) t-+O
t-+O
uniformly for a.e., x E fl, where
In
WI (x)dx =/: 0,
In
w2(x)dx
=/: o.
Theorem 2.17 [LiZol]. Suppose that (wo) holds (1) If Wl(X) ~ 0 for a.e., x E fl, then
(2) If 'l1 2 (X) :::; 0 for a.e., x E fl, then
Cq(I,O)
~
8q,moG, \/q E Z.
where mo is the Morse index of I at 0, 110 Now we can deal with resonant case.
= dim KerI"(O).
Theorem 2.18 [LiZol]. Suppose that (¢~), ('¢'O) hold, (1) If Wl(X) ~ 0,
Shujie Li
250
(2) If W2(X) ::::: 0, dx) 2: 0 for a.e .. x E n. then (1.1) has at least one nontrivial solution. In [LiLiu2] the following conditions were given: (¢l) There exist Cl > 0, 0 < a < 1 such that 1¢(x,t)l::::: Cl(lt!'" (¢2)±
'~~r':)
-+
±oo as
It I -+
+ 1), '
E~, x
En.
uniformly in x E
(x, t)
=
where w(x. t)
=
n, where
lot ¢(x. s)ds. ('l/Jl) There exist C2 > 0, f3 > 1 such that
I'I/J(x, t)1 ::::: c2ltl{3 for It I < 1, x E n, ('l/J2)±
~;r~t)
-+
±oo as
It I -+ 0 uniformly in x E
n.
lot 'I/J(x, s)ds . Theorem 2.19 [LiLiu2]. Suppose that (¢d, ('l/Jd hold, and one of the following conditions was satisfied (1) (¢2)+, ('l/J2)+ hold and Ak =f:. Aj , (2) (¢2)+, ('l/J2)- hold and Ak =f:. Aj+l where we take j such that Aj < Aj+l, (3) «/J2)- , ('l/J2)+ hold and AHl =f:. Aj where we take k such that Ak < Ak+l , (4) (¢2)- , ('l/J2)- hold and Ak =f:. Aj . Then (1.1) has at least one nontrivial solution. If 900 = Ak, assume (¢3)± : There exists T E (1,2) such that
¢(x, t)/IW -+ f3± as t -+
±oo
and
Recently K. Perera and M. Schechter ([PeSc1]) got the following theorem. If (¢3)± holds then leu) satisfies the (PS) condition and (1) Cq(IR,oo) = Oq.mooG provided I(y) > 0 '
251
Some aspects of semilinear elliptic boundary value problem For the case go = Aj, assume ('l/J3)± There exists
(1
E (2,2*) such that
'l/J(x, t)t/IW' -+ O!± as t -+ ±O lo(y)
:=
-.!.( r (1
iy>o
O!+IYI"
+
r
iy
O!_IYI")
i
0 as Y E Ker (- 6 -Aj)\{O}.
Theorem 2.21 [PeSc1]. If ('l/J3)± holds then (1) Cq(J,O) = 8q,mo G provided lo(y) > 0 'Vy E Ker(- 6 -Aj)\{O}. (2) Cq(I,O) = 8q,mo+ vo G provided lo(y) < 0 'Vy E Ker(- 6 -Aj)\{O} . (3) Cq(I,O) = 0 provided lo(y) cahgnes sign on Ker(- 6 -Aj)\{O}. Using Theorem 2.20 and 2.21 It is easy to get the existence of nontrivial solution for (1.1). If local linking condition holds, J . Q. Liu proved the following abstract result. Theorem 2.22 [Liul], Let 0 be an isolated critical point of J The function J E C(X, JR) has a local linking at 0 with (Xl, X2). Then
Ck(I,O)
'1- 0 if k = dimX2.
An important problem arise that the local linking happens with respect to how a direct sum decomposition X = Xl ffiX2. In general we do not know in abstract setting. In some applications, however, the decomposition X = Xl ffi X 2 happened to be such that dimX 2 = mo or dimX 2 = mo + Vo . In [SuI], J. B. Su got the following Theorem 2.23 Then
[SuI]. Unther the assumptions of Theorem 2.22, if J E C 2 . if dimX 2 = mo,
3. Jumping Nonlinear Problems In (1.1) if 9 satisfies (HI)
g(x, t)
lim - -
t-+O+
t
= aI,
. g(x, t) (H2 ) hm - - = t-+oo t
a2,
g(x t) lim ---2.-
t-+O-
t
= dl
uniformly in x E n,
g(x t) lim ---2.- = d2 uniformly in
t-+-oo
t
x
E n,
Shujie Li
252
then we call 9 has jumping nonlinearities at zero if (Hd holds, and at infinity if (H 2 ) holds. First we consider the homogenerous problem
- 6 u = au+ { ulan = 0
+ du-
(3.1)
where u+ = max{u,O}, u- = min{u,O}. From [DaDu3] there exists a continuous function 1)(t) defined on P'l, A2] which satisfies (a) 1) is strictly decreasing, 1)(A2) = A2, lim 1)(A) = +OOj >'-+>'1+ 0
(b) equation (3.1) has no nontrivial solution for (a , d) = (a,1)(a)), a E (AI, A2] and (a, d) = (1)(d), d), dE (AI, A2]j (c) equation (3.1) has no nontrivial solution for Al < d < 1)(a),a E (AI,A2] or Al < a < 1)(d), dE (AI, A2]' We denote by r the curve {(a,1)(a)) : Al < a::; A2} U {(1)(d),d) : Al < d::;
A2} Theorem 3.1 [LiZha3]. Suppose that 9 satisfies (Hd, (H 2),a2,d2 < Al and (al,dd E S:= {(a,d) is above fl . If
Ig(x, ?+- altl -+ ml as t -+ 0+ uniformly in x E S1
It I
Ig(x,t)1+- dltl -+ m2 as t -+ 0 £ormI - 'um y 'm x E H0
It I
(3.2)
<71
(3.3)
<72
where CTI,ml,CT2,m2 > 0 are constants. Then (1.1) has at least one positive solution, one negative solution, one sign-changing solution. Theorem 3.2 [LiZha3]. Suppose that 9 satisfies (Hd,(H2)' al,d l < AI, a2 = d2 E (A;,Ai+d i ~ 2. Then (1.1) has at least one positive solution, one negative solution, one sign-changin solution. Theorem 3.3 [LiZha3] . Suppose that 9 satisfies (HI)' (H2)' (3.2), (3.3), (aI, dd E S, a2 = d2 E (A;,Ai+d,i ~ 2. There exist tl,t2 E IR,tl < 0, t2 > 0 such that g(x, td = g(x , t2) = 0 'r/x E S1. Then (1.1) has at least six nontrivial solutions: two positive solutions, two negative solutions, two sign-changing solutions. Define
2: :={(a,d) E IR2: one called Sk
L:
(3.1) has a nontrivial solution}
the Fucik spectrum of - 6 . Let
= {(a,d) E IR2,Ak :::; a:::; AHl,Ak:::; d:::; Ak+l,
(a, d) :/; (Ak,Ak), (a,d):/; (Ak+l,AHI
Some aspects of semilinear elliptic boundary value problem By [Sch1] we know SK
Dk = {(a,d)
253
f/. L and we define
E]R2 :
there exists a path ret) = (rl(t),r2(t)) such that
(rl(t),r2(t)) f/. Land 1'(0) = (a,d), 1'(1) E Sk}
Theorem 3.4 [LiZha4]. Suppose that 9 satisfies (Hd, (H2), (al,dd E D j , (a2,d2) E D I, j =ll. Then (1.1) has at least one nontrivial solution. Theorem 3.5 [LiZha4] . Suppose that 9 satisfies (Hr), (H2),al,d1 < AI, (a2' d2) E S, (a2,d 2) f/. then (1.1) has at least one positive solution, one negative solution, one sign-changing solution.
L
Theorem 3.6 [LiZha4]. Suppose that 9 satisfies (Hr), (H2)' g(x, 0) = 0 and (H3) (1.1) has a negative strict sub-solution cp(x) and positive strict supersolution 1/;(x), i.e.,
and
- 6 cp > g(cp), cp < 0 { cplao = 0
in
n
- 61/; < g(1/;), 1/;> 0 { 1/;lao = 0
in
n
(al,dr) E D k , (a2,d2) E D j , k,j ~ 2, Then (1.1) has at least seven solutions; two positive solutions, two negative solutions, three sign-changing solutions.
References [AhLP1] S. Ahmad, A. C. Lazer, J. L. Paul, Elementary critical point theory and perturbations of elliptic BVP at resonance. Indiana Univ. Math. J . 25 (1976) 933-944. [AID 1] S. Alama and M. Del. Pino, Solution of elliptic equations with indefinite nonlinearities via Morse theory and linking. Ann. Inst. H. Poincare Anal. Nonlineaire 13 (1996) 95-115. [Ama1] H. Amann, A note on degree theory for gradient mapping, PAMS 85 (1982) 591-597. [AmaZ1] H. Amann, E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear diff. eq., Annali scuola norm . Pisa 7 (1980) 539-603.
254
Shujie Li
[AmI] A. Ambrosetti, Differential equations with multiple solutions and nonlinear functional analysis, Equadiff. 1982; LN in Math. 1017, (1983) 1-22. [AmM1] A. Ambrosetti, G. Mancini, Sharp nonuniqueness results for some nonlinear problems, Nonlinear Anal. 5, (1979) 535-645. [AmR1] A. Ambrosetti, P. Rabinowitz, Dual variational methods in critical point theory and applications, Jour. Funct. Anal. 14 (1973) 349-38l. [BaCW1] T . Bartsch, K. C. Chang, Z. Q. Wang, On the Morse indices of Sign changing solutions of nonlinear elliptic problems, (preprint). [BaLi 1] T . Bartsch, S. J . Li, Critical point theory for asymptotically guadratic functionals and applications to problems with resonance, Nonl. Anal, 7 (1996) 115-13l. [BaWa1] T . Bartsch, Z. Q. Wang, On the the existence of sign changing solutions for semilinear Dirichlet problems, Topological Methods in Nonl. Anal., 7 (1996) 115-13l. [BrNil] H. Brezis. L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math. 64 (1991) 939-963. [BeCDN] H. Berestycki. I. Capuzzo Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Top. Math. Nonlin. Anal. 4 (1994) 59-78. [CNN] A. Castro, J. Cossio, J . Neuberger, Sign changing solutions for a superlinear Dirichlet problem. Rocky Mountain J. Math. 27, (1997) 1041-1053. [Cha1] K. C. Chang, Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure Appl. Math. 34 (1981) 693-712. [Cha2] K. C. Chang, Critical point theory and its applications, Shanghai Sci. Techn. Press (1986), (in Chinese) . [Cha3] K. C. Chang, Variational method and sub-and super-solution. Sci. Sinica, Ser A 26 (1983) 1256-1265. [Cha4] K. C. Chang, Infinite dimensional Morese theory and multiple solution problems, Birkhauser Boston 1993. [Cha5] K. C. Chang, Morse theory in nonlinear analysis. In: A. Ambrosetti. K. C. Chang, I. E. Ekeland (ed), Proc. of Second school of Nonlinear Functional Analysis and applications to Differential Equations, World Scientific (1998) . [ChaLL1] K. C. Chang, S. J . Li, J. Q. Liu, Remark on multiple solutions for asymptotically linear elliptic boundary value problems, Top. Meth. Nonlin. Anal. 3 (1994) 179-187. [Col] D. Costa, A note on unbounded perturbations of linear resonant problems, in Trabalho de Matematica, Vol 245. Univ. Brasillia (1989) . [Dan1] E. N. Dancer, Degenerate critical points, homotopy indices and Morse inequalities, J . Reine Angew. Math. (1984) 350, 1-22.
Some aspects of semilinear elliptic boundary value problem·
255
[DaDu1] E. N. Dancer, Y. H. Du, A note on multiple solutions of some semilinear elliptic problems. J. Math. Anal. Appl. 211 (1997) 626-640. [DaDu2] E . N. Dancer, Y. H. Du, Multiple solutions of some semilinear elliptic equations via the generalized Conley index. J. Math. Anal. Appl. 189, (1995) 848-87l. [DaDu3] E. N. Dancer, Y. H. Du, Existence of changing sign solutions for semilinear problems with jumping nonlinearites at zero. Proc. Royal Soc. Edinburgh, 124A, (1994) 1165-1176. [Hil] N. Hirano, Multiple non-trivial solutions of semilinear ellitic equations, Proc. Amer. Math. Soc. 103 (1988) 468-472. [HLW1] N. Hirano, S. J . Li, Z. Q. Wang, Morse theory without (PS) condition at isolated value and strong resonance problems, Calculus Variations and PDEs, in press. [HoI] H. Hofer, Variational and topological methods in partially ordered Hilbert spaces, Math Ann. 261 (1982) 493-514. [LaSo1] A. C. Lazer, S. Solimini, Nontrivial solutions of operator equations and Morse indices of critical points of Minimax type, Nonlinear Analysis TMA, 12.8 (1988), 761-775. [Lil] S. J . Li , Some new results on Morse theory and Minimax theory, (to appear) . [LiLiu1] S. J . Li, J . Q. Liu, Nontrivial critical point for asymptotically guadratic function, J. Math. Anal. Appl. 165 (1992) 333-345. [LiLiu2] S. J. Li, J . Q. Liu, Computations of critical groups at degenerate critical point and applications to nonlinear differential equations with resonance, Houston J. Math. Vol. 25 , No.3 563-582, 1999. [LiWa1] S. J . Li, Z. Q. Wang, Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems, (to appear). [LiWa2] S. J . Li, Z. Q. Wang, Dirichlet problem of elliptic equations with strong resonances, preprint. [LiWil] S. J. Li, M. Willem, Application of local linking to critical point theory, J . Math. Anal. Appl. 189 (1995) 6-32. [LiWi2] S. J. Li, M. Willem, Multiple solutions for asymptotically linear boundary value problems in which the nonlinearity crosses at least one eigenvalue, Nonlinear Differential Equations and Applications NoDEA, 5 (1998) 479490. [LiZha1] S. J . Li, Z. T. Zhang, Sign-changing solutions and multiple solutions theorems for semilinear elliptic boundary value problems, (to appear) . [LiZha2] S. J. Li, Z. T. Zhang, Multiple solutions theorems for semilinear elliptic boundary value problems with resonance at infinity, Discrete and Continuous Dynamical Systems, Vol 5, No.3, (1999) 489-493. [LiZha3] S. J. Li , Z. T. Zhang, Sign-changing and multiple solutions theorems for
256
Shujie Li
[LiZha4]
[LiZo1]
[Liu1] [Liu2] [LiuLil] [LiuS1] [Ou1]
[PeSc1] [Sch1] [Ra1] [Ra2]
[Si1] [Stl] [SuI] [Til] [Wan 1]
semilinear elliptic boundary value problems with jumping nonlinearities, Acta Mathematica Sinica (New Series), in press. S. J. Li, Z. T. Zhang, Fucik Spectrum and sign-changing and multiple solutions theorems for semilinear elliptic boundary value problems with jumping nonlinearities at zero and infinity, (to appear). S. J . Li, W. M. Zou, The computations of critical groups with an applications to elliptic resonant problems at a higher eigenvalue, J. Math. Anal. Appl 235 (1999) 237-259. J . Q. Liu, Morse index of a saddle point, Syst. Sc. Math. Sc. 2 (1998), 32-39. J . Q. Liu, Sign-changing solutions for elliptic equations of second order, (to appear) . J. Q. Liu and S. J . Li, Some existence theorems on multiple critical points and their applications, Kexue Tongbao 17 (1984), (in Chinese). Z. L. Liu, J. X. Sun, Invariant sets of descending flow in critical point theory with applications to nonlinear differential equations, (to appear). T . Ouyang, On the positive solutions for semilinear equations !:::.u + AU + hu P = 0 on compact manifolds. Part II, Indiana University Math. J. 40 (1992) 1083-1140. K. Perera, M. Schechter, Solutions of nonlinear equations having asymptotic limit at zero and infinity, (to appear). M. Schechter, The Fudk Spectrum, Indiana Univ. Math. J . 43 (4) (1994) 1139-1157. P. Rabinowitz, Variational methods for nonlinear eigenvalue problem, Edicioni cremonese, Roma (1974) 141-145. P. Rabinowitz, Some critical point theorems and applications to semilinear elliptic partical differential equations, Ann. Sculoa Norm. Sup. Pisa, Ser IV. 5 (1978) 215-223. E. A. B. Silva. Critical point theorems and applications to differential equations, PhD Thesis, University of Wisconsin, Madison (1988). M. Struwe, Infinite many critical points for functionals which are not even and applications to nonlinear BVP, Manuscripta Math . 32 (1982) 753-770. J. B. Su, Multiple solutions for semilinear elliptic resonant problems with unbounded nonlinearities, (to appear) . G. Tian, On the mountain pass theorem, Kexue Tongbao 29 (1984) 11501154. (in Chinese). Z. Q. Wang, On a superlinear elliptic equation. Ann. Inst . H. Poincare Analyse Nonlineaire 8 (1991) 43-57.
ITERATION FORMULA FOR THE w INDEX WITH APPLICATIONS Chun-gen Liu Department of Mathematical Nankai University, Tianjin 300071, P.R.China (e-mail: [email protected]) (Dedicated to Professor Paul H. Rabinowitz on his 60th birthday )
Abstract In this paper, we consider various iteration formulae of the w-index. As an application, we consider the prescribed minimal period problems of the second order Hamiltonian systems.
1. Introduction The w index theory for the symplectic path starting from identity was first studied by Y.Long in [Lol]. Now we recall some notation and definition of this theory. We consider linear Hamiltonian systems x = JB(t)x, x E R 2n , (1.1) with B E C(ST' .c s (R 2n)), where ST = R/(rZ) for r > 0, .c(R2n) denotes the set of 2n x 2n real matrices, and-.c s (R2n) denotes its subset of symmetric ones. Z, N, Rand C denote the set of integer numbers, natural numbers, real numbers and complex numbers, respectively. It is well known that the fundamental solution 'Y(t) of (1.1) is a path in the symplectic group Sp(2n) = {M E .c(R2n )IM T JM = J} with J =
(~ ~I),
I is the identity on R 2n , i.e. l' E P (2n) with
PT(2n)
T
=b
For any l' E P(2n) and w E U to a pair of integers
E C([0,r],Sp(2n))I'Y(0)
= I}).
= {z E q Izl = I}, the w index of l' is defined
(iT,W (')'), vr,w (1')) E Z x {O, 1, ... ,2n}. 257
Chun-gen Liu
258
If w = 1, this index is called the Maslov-type index and denoted by (iT (-y) , VT(-y))' The iteration property of the Maslov-type index was studied deeply in [DL] , [LLl], [L02] and [LZh]. The iterated Maslov-type index theory deals with the index (ikT( . yk),vkT(-l)) with kEN, where,k is defined by ,k(t) = ,(t - j,),(,)i, 'Vj,:::; t :::; (j
+ I)"
j = 0,1" " , k - 1.
In [DL] and [LLl],the iteration index (ikT(-yk),VkT(-yk)) was estimated by the original index (iT (-Y), vT(-y)) or the mean Maslov-type index. In [L02] and [LZh], the precise iteration formula was studied. The iteration theory of the Maslov-type index has taken an important role in the studying of the nonlinear Hamiltonian systems(see [DL], [LLl], [LL2], [L02] and [LZh]). In this paper, we consider the iteration formula for the w index, i.e., we consider the iterated properties of the index (ikT,w(-yk), VkT ,w(,k)). Specially we consider precise iteration formula for the case of w = -1. As an application, we consider the minimal periodic problem for the second order Hamiltonian systems. More precisely, We consider the existence of non-constant periodic solutions with prescribed minimal period for the following autonomous second order Hamiltonian systems (1.2) x + V'(x) = 0, 'Vx E R 2n , where n E N.V : R 2 n -+ R is a function, and V' denotes its gradient. In this paper, we denote by a· band lal the usual inner product and norm in R 2n , and (-,.) the Hermitian inner product in C. We assume that V satisfies the following conditions (VI) V E C2(R2n,R). (V2) There exist constants J.l
> 2 and
TO
> 0 such that
0< J.lV(x) :::; V'(x) . x, 'Vlxl ~
TO.
= 0,
(V3) V(x) ~ V(O)
'Vx E R2n. (V4) V(x) = o(lxI ) as Ixl -+ O. (V5) V is even, i.e. V( -x) = V(x), 'Vx E R2n. (V6) There exist constants w > 0 and Tl > 0 such that 2
In his pioneering work [Ral] of 1978, Rabinowitz proved that, for any, > 0, the system (1.2) possesses a non-constant ,-periodic solution under the conditions (Vl)-(V4). Under a similar conditions(replaced V by the function H defined
Iteration formula for the w index with applications
259
on R2n, i.e. conditions (Hl)-(H4)), he also proved that the existence result is true for the first order Hamiltonian systems i = JIJ'(z), z E R2n,
(1.3)
Because of a T / k-periodic solution is also a T-periodic solution for every kEN, Rabinowitz conjectured that (1.2) or (1.3) possesses a non-constant solution with any prescribed minimal period under his conditions. Since then, a large amount of contributions on this minimal period problem have been made by many mathematicians (cf. [AZ], [AC], [AM], [CE], [DL], [EH], [Ek], [FQ], [FW], [GMl], [GM2], [GM3], [LLl], [Lo3], [Lo4J) . Among these work, a significant progress was made by Ekeland and Hofer in [EH] in 1985. They gave an affirmative answer to Rabinowitz' conjecture for strictly convex Hamiltonian systems (1.3). Their work was extented to the case of system (1.2) when V is strictly convex by Coti Zelati, Ekeland, and P.L. Lions(cf. Theorem IV.5.3 of [EkJ). Generalizations of their results under different or weaker convexity assumptions can be found in [Ek], [GMI-GM3] and [LLl]. Most of them assumed some kinds of convexity conditions on the function V or H. As far as the author knows, there are only four papers ([GMl], [GM2], [Lo3] and [Lo4J) dealing with the functions with no convexity assumptions. In [GMl] and [GM2], by an a priori estimation method Giradi and Matzeu considered the minimal period problems of the first order Hamiltonian systems under the conditions (Hl)-(H4)(these conditions hold globally on R2n) and some additional conditions. In [Lo4], Y. Long proved that, under conditions (Vl)-(V4), for every T > 0 there exists an even T-periodic solution of (1.2) with minimal period not smaller than T/(n+2). In [Lo3], under conditions (Vl)-(V5), it was proved that for every T > 0, there exists an even T-periodic solution of (1.2) with minimal period T or T/3) .
2. Iteration formula of the w index theory 'In this section, we shall study the iteration formula for the w index theory. We start with the following result. Lemma 2.1. For any 'Y E Pr(2n), there holds
Particularly, if k = 2m - 1 is odd, and w = -1, then by the symmetry of windex on the unit circle of the complex plan, i.e. if/,r = i1),r, where fj = e-v'=I9 if1J = ev'=I9 , it implies that LI,kT and i-l,r have the same parity.
Chun-gen Liu
260
Proof. This Lemma can be proved directly by the similar arguments of Lemma 1.5.2 and Corollary 1.5.4 of [Ek] and the saddle reduction method(see [LoI]).
o For ME Sp(2n) and wE U, denote by S!(w) the splitting numbers defined by Theorem 1.3 of [LoI] . We define the functions E and [ ] : R -+ Z by
E(x)
= min{k E ZI k ~ x},
[x]=max{kEZlk::;x}, VxER. We now state the main result of this section. Theorem 2.2. For any, E P T (2n), we put M kEN , there holds
imT,-l (,m)
= m(i
T
,-l(r)+s14( -l))-S14( -1)+2
= ,(7) .
For m
= 2k -
1,
L [-m:7r+ 7r] SM(e...f-fO),
BE(-1T,1T)
(2.2)
= 2k,
for m
kEN, there holds
imT,_l(rm)
= m(iT,-l(r) + S11(-I)) + 2
L
[-m: + 7r] SM(ev'=IB),
7r
BE( -1T ,1T)
(2.3) Proof. We follow the ideas in [LZh] to prove the Theorem. We only prove (2.2). The proof of (2.3) is similar and omitted. By (2 .1), there holds
imT,_l(rm)
=
L
iT,w(r) = iT,-l(r)
w7n=_l
'=~'T' (i'._. -.,,'f
+ (0) + ',!'" St/ (eyCT,) = iT,-l(r) + (m -l)(i T,-l(r) + S11(-I))
+
'EE. )CW~,-;,
St/(eyCT,) -
-gt.~,-;,
= iT,-l(r) + (m -I)(iT,-l(r) + S11(-I)) + L 1 _ [m~: 7r]) S11(ev'=I8)
(m; (m; 1_E(m~: 7r) + 1)
8E(-1T ,1T)
L 8E(-1T,1T)
_'<'~'!;'"
SM(ev'=I8).
SM(e yCT'))
SM(eyCT'))
Iteration formula for the w index with applications
-mO + 1T -21T- + mO21T+ 1T
there holds
and
We have
=
1 ,
261
Chun-gen Liu
262
This implies that imr,_l(-rm) = ir,-l(-r)
+ (m -1)(ir,-l(-r)
+St(eV'=I9)) - 2
SM(HO)
L
9E(0,1T)
L L
+
2E
9E(0,1T)
+
(m~: 7r) SM(eV'=IO)
(-2 + 2E(m~: 7r))
SM(eV'=I9)
9E( -1T,0)
ir,-l(-r) +2
+ (m -
L
E
l)(i r ,-l(-r)
+ St(eV'=I9))
(m~: 7r) SM(eV'=I9) -
2
9E(-1T,1T)
=
L
SM(eV'=I9)
9E(-1T,1T)
ir,-l(-r) + (m -1)(i r ,-l(-r) + St(eV'=I9)) +2
L [-m:7r+ 7r] SM(eV'=I9) .
9E(-1T,1T)
This complete the proof.
o
3. Symmetric variational structure For r
> 0, let
Sr = R/(rZ) and Er = Wl,2(SnR2n) with the usual norm
Then Er is a Hilbert space. We denote by (., ·)r the corresponding inner product in E r . Now we define the 'odd' subspace E~dd of Er by E~dd
= {u E Er Iu(r/2 + t) = -u(t)}.
It is clear that E~dd is also a Hilbert space with equivalent norm
By (V5), the function V is even, we define a functional
f on
E~dd
by
(3.1)
Iteration formula for the w index with applications
263
It is clear that any critical point x of f in E~dd is a solution of the system (1.2) with x(T/2 + t) = -x(t) . If x E E~dd is a critical point of f in E~dd, then the following quadratical form defined in E~dd by (3.2)
has finite Morse index and nullity which are defined as the dimension of the greatest negative definite subspace and the null subspace, respectively. By (V5), A(t) := V"(x(t)) is T/2-periodic, we can see that there holds
r/ (lul 2
'l/Jr(u,u)
=2
10
2
-
(V"(x(t))u,u))dt, Vu E E r / 2,-1,
(3.3)
where E r / 2,-1 = {u E Wl,2([o,T],Rn) I u(T/2) = -u(O)}, thus by the arguments of [Lol], we know that in fact the Morse index pair (m-(x), v(x)) is exactly the w (w = -1) index (L 1,r/2(X), V-l,r/2(X)) of x as a solution of the second Hamiltonian system which correspending to a linear first Haimiltonian system with coefficent matrix B(t) =
(~ ~(t))'
By the mountain pass lemma, we have the following result. Theorem 3.1. Suppose the function V satisfies the conditions (Vl)-(V5), then for every T > 0, there is a critical point x of the functional f in E~dd with its Morse index satisfying
(3.4) If the function V satisfies the conditions (Vl)-(V3), (V5)-(V6), then for every there is a critical point x of the functional f in E~dd with its Morse index satisfying
o < T < 27r /,;w,
(3.5) Proof. We only prove the first result, the second result is similar to prove. It is easy to check that the function f in E~dd satisfies all conditions of the mountain pass lemma. In fact, by the condition (V4), it is easy to see that there exist p > 0 and a > 0 such that f(u) ~ a
> 0,
Vu E E~dd
n Bp(O).
On other hand, f(O) = 0 and by (V2), there exists R > p and Uo E E~dd with Iluoll ~ R such that f(uo) ~ O. Thus by mountain pass lemma, there exists a critical point x with its morse index m - (x) ~ 1.
o
Chun-gen Liu
264
If x E E~dd is a non-zero solution of the system (1.2), set y(t) = x(t) and = (x(t),y(t))T, then z satisfies z(7/2+t) = -z(t) for all t E R and it is a non-zero 7-periodic solution of the following first order Hamiltonian system
z (t)
z(t) where H(z)
= 1/21Y12 + V(x).
= JH'( z (t))
(3.6)
From (3.6), there holds
z.. - JH"( z )z.. Thus
z is a non-zero solution of the following linearized equation u = JH"( z )u, u E R2n .
(3.7)
Let TZ(t) be the fundamental solution starting from the identity matrix hn . Then -1 E a(')'z(7/2)), thus we have
(3.8) On other hand , if x E E~dd is a non-zero solution of the system (1.2), then it is not even times iteration of any periodic solution of (1.2) . i.e., if x is a 7/k-periodic solution of (1.2), then k is odd. Further more, x E E~1~, i.e., x(7/(2k) + t) = -x(t) . In fact, suppose k = 2m - 1, mEN, then there holds 7
x(2(2m _ 1)
+ t)
7 = x( 2(2m _ 1)
(m - 1)7 1)
+ 2(2m _
+ t)
7 = x("2
+ t)
= -x(t). (3.9)
4. Iteration inequality of the symmetric Morse index theory and the minimal periodic problems Let x E E~dd be a critical point of the functional f on E~dd, and A(t) = V"(x(t)). By the Proposition 2.3 of [Lo1], f(x) defines the following bilinear form on E~dd
1/Jr(x,y)
r (x ·iJ-A(t)x·y)dt = ior/ (x ·iJ-A(t)x · y)dt,
= io
2
2
(4.1) We denote the Morse index of'l/J on E~dd by (L 1,r/2(X),1I-1 ,r/2(X)) . For odd number k E 2N - 1, by the arguments in the end of the section 3, the k-times iteration of the solution x is a kT-periodic solution of (1.2) satisfies x(kT /2 + t) = -x(t) . So we can define the iterated index (i-l,kT/2(X),1I-1,kT/2(X)). Estimating the iterated index is crucial in our work here.
Iteration f(j)Fmula for the w index with applications
265
We note that if setting u(t) = x(t) and v(t) = y(t) in (4.1), then x(t) = (II T u)(t) and y(t) = (II T v)(t), the operator lIT : L~ --+ W 1,2([0, T], Rn) is bounded and defined in [Ek] . Thus we can define the following quadratic form in L~ = L2([0, T], R2n) (4 .2)
Defining an operator AT : L~ --+ L~ by
there holds (4.3)
where we have denoted £T = I - AT and (,,·h the L2 inner product. For the iterated symmetric Morse index, we have following estimate. Theorem 4.1. Suppose A(t) ~ O. For k E 2N - 1, there holds . k-l(. Ll,kT/2 ~ -2- Ll,T/2
+ /J-l,T/2 ) .
(4.4)
Proof. The case k = 1 is trivial. In the following we suppose k > 1. Firstly we note that the Morse indices of ¢>T in L~ equal to the Morse indices of 'l/JT in the space E~dd, and there is a ¢>T-orthogonal splitting
with: dimL_
= L 1 ,T/2 and ¢>T(U,U) < 0 Vu E L_ \ {O} dimL o = /J-l,T/2 and Lo = ker¢>T'
For 0 ~ j ~ (k - 1)/2 and k > 1, we define an operator T j
Tju(t) = {
:
U(t-jT) jT~t~jT+T/2 . - (k-l)T)' kT < t < . U(t, - JT 2 JT + "2 _ _ JT o otherwise
Now define subspaces M j and N j of L~T by
L~ --+ L~T by
+ (k+l)T 2
Chun-gen Liu
266 The subspaces are mutually orthogonal. Setting M = ffiMj and
N = ffiN j , j
j
we check easily that
k-1
dim M ED N = -2-(L l :,r/2 For W
E
+ 1I-1,r/2)'
Ihl
small, the operator I + h£kr : L%r -+ L%r is an isomorphism. For M ED N with IIwII2 = 1, setting
then we have
:h
= (£kT W, :hgh(w)lh=O ) = (£kTW, £kTW)2 - (£kTW, w)~ = II£krwlI~ -
We note that N n ker
Since the unit sphere M ED N n 8 1 of M ED N is compact, it follows that there is an 10 > 0 such that, whenever -10 < h < 0, we have
So
o Corollary 4.2. Suppose that V satisfies (V1)-(V5) and VII (x) is semi-positive definite. Then for every T > 0 the system (1 .2) possesses a non-constant Tperiodic solution with minimal period T . Proof. By Theorem 3.1, there exists a non-constant solution x E E~dd with its index satisfying
(4.5)
Iteration formula for the w index with applications
267
Suppose its minimal period is T/k, then k is an odd number, and by (3.8) there holds (4.6) 1I-1 ,T/(2k) ~ 1. Thus by Theorem 3.2, if L
1 ,T/2
= 0, there holds
k-l
-2-(i-l ,T/(2k)
+ 1I-1 ,T/(2k»)
(4.7)
:::; O.
Since L 1 ,T/(2k) ~ 0, from (4.7), we have k = 1. If i-l,T/2 = 1, then by Lemma 2.1, L 1 ,T/(2k) = 1. Thus there holds
k-l
k - 1 :::; -2-(L 1 ,T/(2k)
+ 1I-1,T/(2k»)
:::; 1.
(4.8)
From (4.8), it follows k :::; 2, but k is an odd number, so k = 1.
o Corollary 4.3. Suppose that V satisfies (V1}-(V3), (V5},(V6) and V"(x) is semi-positive definite . Then, for every 0 < T < 27r/VW the system (1 .2) possesses a non-constant T-periodic solution with minimal period To Proof. By the second part of Theorem 3.1 and the same arguments as above, we can prove this result.
o With no convexity assumption , we have the following results. Theorem 4.4. Suppose that V satisfies (V1}-(V5) and n = 1. Then for every T > 0 the system {1.2} possesses a non-constant T-periodic solution with minimal period T. Suppose that V satisfies (V1}-(V3), (V5},(V6). Then, for every 0 < T < 27r/VW the system {1.2} possesses a non-constant T-periodic solution with minimal period T. Proof. We continue the proof of Theorem 4.2. For the T-periodic solution x of (1.2), by (3.8), there holds
'YX(-;k)",(~1 ~1)'
b=O,±I,
where A '" B means the matrices A and B are homotopic in the sense of [Lol) . Now we divide our proof into three cases. case 1. 'YX(-{k) '"
(~1 ~1).
For this case there holds St(-I)
= 1,
then by Theorem 2.2, we have i T / 2 ,-1
= k(iT/(2k),-1 + 1) -
1.
By (4.5) and i T/(2k),-1 ~ 0, there holds k :::; 2. It implies k = 1 by that k is odd .
Chun-gen Liu
268
case 2. 'YX(-{k) .....
(~1 ~1).
For this case there holds St(-I)
=0
and i T / 2 ,-1 is odd, then by Theorem 2.2, we have iT /2,-1 = kiT /(2k),-1·
Thus by (4.5) and i T /(2k),-1 is also odd, we have k = l. case 3. 'YX(-{k).....
( -1 -1) 0
-1
+ . For this case there holds SM(-I)
= 1,
by Theorem 2.2, there holds i T / 2 ,-1 = k(i T /(2k),-1
So we have k
+ 1) -
1.
= 1 by the argument of the case 1. o
Corollary 4.5. If V : Rn -+ R satisfies
and each Vj(Xj)' j = 1,···, n satisfies the condition (V1)-(V5) , then for every 7 > 0, the system (1 .2) possesses a non-constant 7-periodic solution with minimal period 7 .
References [AZ] H. Amann and Zehnder, Nontrivial solutions for a class of nonresonace problems and applications to nonlinear differential equations. Ann. Scuola Norm . Super. Pisa,7(1980) 539-603. [AC] A. Ambrosetti and V. Coti Zelati, Solutions with minimal period for Hamiltonian systems in a potential well Ann.I.H.P., Anal. non lineaire. 4 (1987) 275-296. [AM] A. Ambrosetti and G. Mancini, Solutions of minimal period for a class of convex Hamiltonian systems math. Ann. 255 ( 1981 ) 405-421. [CE] F. Clarke and I. Ekeland , Hamiltonian trajectories having prescribed minimal period .Comm. pure Appl. Math. 33 (1980)103-116. [DL] D. Dong and Y. Long, The iteration formula of the Maslov-type index theory with applications to nonlinear Hamiltonian systems. Trans. Amer. Math. Soc. 349(1997)2619-2661. [Ek] I. Ekeland, Convexity Method in Hamiltonian Mechanics. Springer-Verlag. Berlin(1990).
Iteration formula for the w index with applications
269
[EH] I. Ekeland and H. Hofer, Periodic solutions with prescribed period for convex autonomous Hamiltonian systems. Invent. Math . 81 (1985)155188. [FQ] G. Fei and Q. Qiu, Minimal periodic solutions of nonlinear Hamiltonian systems. Nonlinear Analysis T.M.A . 27 (1996)821-839. [FW] G. Fei and T. Wang, The minimal period problem for nonconvex even second order Hamiltonian systems. J. Math. Anal. Appl. 215 (1997)543559. [GM1] M. Giradi and M. Matzeu, Some results on solution of minimal period to superquadratic Hamiltonian equations. Nonlinear Analysis T.M.A . 7 (1983) 475-482. [GM2] M. Giradi and M. Matzeu, Solution of minimal period for a class of nonconvex Hamiltonian systems and applications to the fixed energy problem. Nonlinear Analysis T.M.A. 10 (1986) 371-382. [GM3] M. Giradi and M. Matzeu, Dual Morse index estimates for periodic solutions of Hamiltonian systems in some nonconvex superquadratic case. Nonlinear Analysis T.M.A . 17 (1991)481-497. [LL1] C. Liu and Y. Long, Iteration inequalities of the Maslov-type index theory with applications. Nankai Inst. of Math. Preprint Series. No. 1997-M014. Revised 1998. to appear in Journal of Differential Equations. [LL2] C. Liu and Y. Long, Hyperbolic characteristics on star-shaped hypersurfaces. Nankai Inst. of Math. Preprint Series. No. 1997-M-01O. Revised 1998. to appear in Ann. l. H. P Ana. non lineaire. [Lo1] Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math ., 187(1999)113-149. [L02] Y. Long, Precise iteration formula of the Maslov-type index theory and ellipticity of closed characteristics, Nankai Inst. of Math. Preprint Series. No. 1999-M-001. [L03] Y. Long, The minimal period problem of classical Hamiltonian systems with even potentials. Ann. I.H.P. Anal. non lineaire. 10 (1993)605-626. [L04] Y. Long, The minimal period problem of periodic solutions for autonomous superquadratic second order Hamiltonian systems. J. Diff. Eq. 111 (1994) 147-174. [LZh] Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in R2n. Nankai Inst. of Math. Preprint Series. No. 1999-M-002. [Ral] P. Rabinowitz, Periodic solutions of Hamiltonian systems. Comm. pure Appl. Math.31 (1978) 157-184.
270
Chun-gen Liu
[Ra2] P . Rabinowitz , On the existence of periodic solutions for a class of symmetric Hamiltonian systems. Nonlinear analysis T.M.A. 11 (1987) 599611.
Dynamics on compact convex hypersurfaces in
R 2n Yiming Long*t Nankai Institute of Mathematics, Nankai University Tianjin 300071, The People's Republic of China Dedicated to Professor Paul Rabinowitz on his 60th birthday
1
Introduction
Let ~ be a C 2 -compact hypersurface in R 2 n bounding a convex compact set C with non-empty interior, and ~ has a non-vanishing Gaussian curvature. Without loss of generality, we assume 0 E C. We denote the set of all such hypersurfaces in R 2 n by 1i(2n). We denote by 1is(2n) the subset of 1i(2n) which consists of all hypersurfaces symmetric to the origin, i.e. x E ~ implies -x E ~. For x E ~,let Nr:.(x) be the outward normal unit vector at x of~. We consider the dynamical problem of finding r > 0 and an absolutely continuous curve x: [0, r] -t R 2 n such that
x(t) { x(r) where J
= (~ ~I)
= JNE(x(t)), =x(O),
x(t) E
~,
\:It E R,
(1.1)
is the standard symplectic matrix on R 2n . A solution
(r, x) of the problem (1.1) is called a closed characteristic on ~. Two closed characteristics (r,x) and (a,y) are geometrically distinct, if x(R) -::p y(R). We denote by ..J(~) and j(~) the set of all closed characteristics (r,x) on ~ with r being the minimal period of x and the set of all geometrically distinct ones respectively. For (r, x) E ..J(~), we denote by [(r, x)] the set of all elements in ..J(~) which are geometrically the same as (r,x). Denote by 1i-(2n) and ·Partially supported by the NNSF and MCME of China, the Hong Kong Qiu Shi Sci . Tech. Foundation, and CEC of Tianjin. t Associate member of ICTP.
271
Yiming Long
272
1iOO(2n) the sets of all ~ E 1i(2n) with # j(~) < +00 and # j(~) = +00 respectively, and by H; (2n) the intersection of 1is(2n) and 1i- (2n) . Fix a ~ E 1i(2n) bounding a convex set C with the origin being in the interior of C . Let jc : R 2n --+ [0, +00) be the gauge function of C defined by jc(O)
=0
and
jc(x) = inf{)'
> 01 ~ E C} for
x:j:. O.
(1.2)
Fix a constant a satisfying 1 < a < 2 in this paper. As usual we define the Hamiltonian function HOI : R 2 n --+ [0, +00) by (1.3) Then HOI E C 1 (R2n,R) n C 2 (R2n \ {O},R) is convex and ~ = H;;I(I). It is well-known that the problem (1.1) is equivalent to the following given energy problem of the Hamiltonian system
x(t) { X(7)
= JH~(x(t)), = x(O) .
"It E R,
HOI(x(t)) = 1,
(1.4)
Denote by .J(~, a) the set of all solutions (7, x) of the problem (1.3) where 7 is the minimal period of x . Note that elements in .J(~) and .J(~, a) are one to one correspondent to each other. Let (7,X) E .J(~,a). The fundamental solution 'Yx: [0,7)--+ Sp(2n) with 'Yx (0) = I of the linearized Hamiltonian system
y(t) = JH~(x(t))y(t),
\It E R,
(1.5)
is called the associated symplectic path of (7, x) . The eigenvalues of 'Yx(7) are called Floquet multipliers of (7, x). By Proposition 1.6.13 of [10], the Floquet multipliers with their multiplicity and Krein signs of (7, x) E .J(~) do not depend on the particular choice of the Hamiltonian function in (1.4) . For any M E Sp(2n), we define the elliptic height e(M) of M to be the total algebraic multiplicity of all eigenvalues of M on the unit circle U = {z E c Ilzl = 1} in the complex plane C. Since M is symplectic, e(M) is even, and o ~ e(M) ~ 2n. As usual a (7, x) E .J(~) is elliptic, if e{'Yx(7)) = 2n. It is nondegenerate, if 1 is a double Floquet multiplier of it. It is hyperbolic, if 1 is a double Flquet multiplier of it, and e{'Yx(7)) = 2. It is well known that these concepts are independent from the choice of a > 1. Since ~ E 1i(2n) is compact, the flow of (1.1) is always defined all the time. But it is completely nontrivial to say more about the periodic boundary value problem (1.1). A typical example of ~ E 1is(2n) is the ellipsoid En(r) defined as follows . Let r = (TI, .. . ,Tn) with Tk > 0 for 1 ~ k ~ n. Define En (T)
= { X = (Xl, ... ,Xn )
ER
2n
IXk 12 1-21 ~ L - 2 - = I}. r k=l
k
(1.6)
Dynamics on compact convex hypersurfaces in R 2n
273
If rj/rk is irrational whenever j ::/= k, this En(r) is called a weakly nonresonant ellipsoid. In this case it is easy to verify that # j(En(r)) = n, that all the closed characteristics on En(r) are elliptic and nondegenerate, and that all of them have irrational mean indices (d. §1.7 of [10]). My aim in this paper is to give a survey on the study of the existence, multiplicity, and stability problem of (1.1). I shall also propose some possible further considerations and open problems in this field.
2 2.1
Known results Existence and multiplicity
The most famous long standing conjecture on (1.1) is (d. p.235 of [10]): \II; E 1I.(2n) .
(2.1)
It seems that the first study on this problem started by A. M. Liapunov in [20] of 1892 and J . Horn in [19] of 1903. It is surprising enough that more than one hundred years ago, they were able to prove the following great result for (1.1) in the local sense: Suppose H : R 2 n ~ R is analytic, a(JHI/(O)) = {±Awl, ... , AWn} are purely imaginary, and satisfy ~ (j. Z for all i ::/= j. Then there exists fO > 0 small such that )
\10
< f < fO .
(2.2)
This deep result was greatly improved by A. Weinstein in [41] of 1973. He was able to prove that for H E C 2(R2n,R), if HI/(O) is positive definite, then there exists fO > 0 small such that (2.2) still holds. Weinstein's this theorem then was further generalized by J . Moser in [36] of 1976 to more general systems and by T. Bartsch in [2] of 1997 to non-convex case for the local problem. In the full generality of (2.1), i.e. the problem (1.1) in the global sense, the excited breakthrough was made by P. Rabinowitz in [37] (for I; being starshapted) and A. Weinstein in [42] of 1978. They proved \II; E 1I.(2n) .
(2.3)
Here in his pioneering work, P. Rabinowitz first introduced the variational method to study the problem (1.1), which stimulated great interests and many papers on this problem. Among the great amount of papers on this problem, 1. Ekeland introduced his Morse-type index theory for convex Hamiltonian systems, and made profound studies on the problem (1.1). 1. Ekeland-L. Lassoued in [13] and 1.
Yiming Long
274
Ekeland-H.Hofer in [11] of 1987, and A. Szulkin in [39] of 1988 proved the following multiplicity result: #
jCY:,)
~ 2,
'V~ E 1l(2n)
with n
~ 2.
(2.4)
Because it is so difficult to completely anwser the conjecture (2.1), It is natural to study the multiplicity of closed characteristics on special subclasses of 1l(2n) under additional conditions. Following studies for closed geodesics on Riemannian manifolds, the pinching condition is introduced to the study of problem (1.1). We say ~ E 1l(2n) is b-pinched, if there exist two balls Br and BR with radius rand R respectively satisfying Br C C C BR and R/r < b, where C is the convex domain bounded by ~. In [12] of 1980, I. Ekeland and J.M. Lasry proved if ~ E 1l(2n) is V2-pinched.
(2.5)
A different proof of this result can be found in [17] of H. Hofer of 1982. In [1] of 1981, A. Ambrosseti and G. Mancini further proved if ~ E 1l(2n) is v'k-pinched,
(2.6)
where [a] = max{k E Z I k ~ a} for any a E R. When ~ is symmetric to the origin, better result was obtained by M. Girardi In [15] of 1984, where he proved if ~ E lls(2n) is V3-pinched.
(2.7)
In [3] of 1985, H. Berestycki, J. M. Lasry, G. Mancini, and B. Ruf generalized (2.5) to (star-shaped hypersurfaces under) pinching conditions in terms of ellipsoids. Using a theorem of J . Franks, in [18] of 1998, H. Hofer, K. Wysocki, and E. Zehnder further improved (2.3) with n = 2 to # j(~)
=2
or
+ 00,
'V~ E
1£(4).
(2.8)
Another interesting result proved by I. Ekeland in [9] of 1986 and [10] of 1989 is the following: # j(~) = +00 holds generically for ~ E 1l(2n) in CPtopology with 3 ~ p ~ +00.
2.2
Stability
We are interested in the linear stability of closed characteristic (r, x) on a given ~ E 1l(2n). Here such a (r, x) is linearly stable, if there exists a constant c > 0 such that there holds 'Vk E N.
(2.9)
Dynamics on compact convex hypersurfaces in R 2n
275
Then it is easy to see that (7, X) is linearly stable if and only if it is elliptic. For any E E 1i(2"!.), we denote by je(E) or jh(E) the set of all elliptic or hyperbolic elements in J(E) respectively. On this stability problem, the following is the most important long standing conjecture (cf. p.235 of [10]): # -
Je(E) 2': 1,
'IE E 1/.(2n).
(2.10)
So far there is no much progress on this conjecture. The first result in this direction seems to be that proved by I. Ekeland in [9] of 1986: if E E 1/.(2n) is V2-pinched.
(2.11)
In [9] of 1. Ekeland in 1986 and [27] of myself in 1998, the following alternative result for any E E 1/.(2n) was proved: either there exist infintely many geometrically distinct hyperbolic closed characteristics [(7j, X j)] for j E Nwith their minimal periods 7j -+ +00, or jh(E) -:j; j(E). Specially, this implies that (2.12) The following remarkable result was proved by B. D'Onofrio, G. F. Dell'Antonio, and 1. Ekeland in [6] of 1992, when E is symmetric. (2.13)
3
Recent progress
Based upon the iteration theory of the Maslov-type index established in recent several years by [29] and [31], in my paper [31], [35] of C. Zhu and myself, and [21] of C. Liu, C. Zhu, and myself, we obtained some new progresses in this field . Let N and Z denote the sets of natural numbers and integers respectively.
Definition 3.1 ([35]) Fora E (1,2), we define a map l!n: 1/.(2n) -+ NU{+oo} by en(E)
~{
+00, if #V(E, a) = +00, min{[i(X,1)+2S+(;l-v(X,1)+n]I [(7, x)] E Voo(E, a)}, if #V(E, a) < +00.
(3.1)
Here (i(x, 1), v(x, 1)) is the Maslov-type index of (7, x), S+(x) is the splitting number of (7,X). They and the concepts V(E,a) and Voo(E,a) are explained below. As proved in [35], l!n(E) does not depend on the choice of a E (1,2) and is a shape invariant, i.e. is independent of dilations of E. Our main multiplicity and stability results are the following theorems.
Yiming Long
276 Theorem 3.1 ([35]) There holds
2:
# j(,}:,)
€?n(~) '
\f~
E H(2n).
(3.2)
By this theorem, to get the lower bound on multiplicity of closed characteristics on ~ E H(2n), it suffices to estimate €?n(~) from below. Theorem 3.2 ([35]) There holds n
2:
€?n (~)
n
2: ["2 J + 1,
~
E 1r(2n) .
(3.3)
Theorem 3.3 ([35]) Fix ~ E H(2n) and 0: E (1,2) . Suppose every (T,X) E .J(~, 0:) satisfies , (3.4) i (x , 1) + 2S+(x) - v(x, 1) 2: n. Then there holds
(3.5) In particular, when every (T,X) E .J(~) is nondegenerate, (3.4) must hold, and then (3 .6)
When
~
is symmetric to the origin, better result holds.
Theorem 3.4 ([21]) There holds, €?n(~)
2: n ,
\f~ E
(3.7)
Hs(2n) .
On the stability conjecture (2 .10), we are able to prove the following results: Theorem 3.5 ([35]) There holds # .Je(~)
2:
1,
(3.8)
Theorem 3.6 ([35]) On any ~ E H-(2n) , there exist at least €?n(~) - 1 elements in j(~) which possess irrational Maslov-type mean indices. Theorem 3.7 ([31]) There holds je(~)
= j(~),
(3.9)
By (2.8), this result is further improved by the following Theorem 3.8 ([35]) For any ~ E H- (2n) with n there holds
2: 2,
if# j(~) ::; 2€?n(~) -2, (3.10)
Dynamics on compact convex hypersurfaces in R 2n
4
277
Main new ideas
In the following, we briefly explain the notations appeared in and main ideas of Theorems 3.1-3.8.
4.1
Maslov-type index functions and splitting numbers
As usual, the symplectic group Sp(2n) is defined by Sp(2n)
= {M E GL(2n, R) I MT J M = J},
whose topology is the one induced from that of paths in Sp(2n):
2
R4n
•
We are interested in
P T (2n) = bE C([O, 7J, Sp(2n)) h(O) = I}, which is equipped with the topology induced from that of Sp(2n) . The following real function is defined in [28]:
Dw(M)
= (_I)n- 1wn det(M -
wI),
Vw E U, M E Sp(2n).
Thus for any w E U, the set Sp(2n)~ = {M E Sp(2n) I Dw(M) = O} is a co dimension 1 hypersurface in Sp(2n). For any M E Sp(2n)~, we define the orientation of Sp(2n)~ at M by the positive direction ftM etdlt=o of the path M etfJ with 0 ::; t ::; 1 and € > 0 being sufficiently small. We also define Sp(2n)~ = Sp(2n) \ Sp(2n)~. For any two continuous arcs ~ and 11 : [0,7] -+ Sp(2n) with ~(7) = 11(0), we define as usual:
11 * ~(t) = {
~(2t),
11(2t - 7),
if 0 ::; t ::; 7/2, if 7/2 ::; t ::; 7.
We define a special path ~n E P T (2n) by ~n(t)
. t t t)_l , ... , (2 - -t)_l) , = dlag(2 - -, ... ,2 - -, (2 - 7
7
7
for 0 ::; t ::;
7.
7
Definition 4.1 Let wE U. For any M E Sp(2n) , we define
lIw(M) = dime kerc(M - wI) . For any
7
>
(4.1)
0 and, E PT(2n), we define
lIw(r) = lIw(r(7)).
(4.2)
If ,(7) E Sp(2n)~, we say, is w-nondegenerate and define
iw(r) = [Sp(2n)~ :, * ~nJ,
(4.3)
278
Yiming Long
where the right hand side of (4 .3) is the usual homotopy intersection number, and the orientation of'Y * ~n is its positive time direction under homotopy with fixed end points. If 'Y(r) E Sp(2n)~, we define (3 E P,.(2n) is w-nondegenerate
(4.4)
and CO -sufficiently close to 'Y}' Then (iwb),vwb)) E Z x {0,1, .. . ,2n},
is called the index function of'Y at w . Note that the right hand side of (4.4) is always finite as proved in [29] or [30] respectively. For any symplectic path 'Y E p,.(2n) and mEN, we define its m-th iteration 'Y m : [0, mr] -+ Sp(2n) by 'Ym(t)
= 'Y(t -
'Vjr ~ t ~ (j
jrh(r)j,
+ l)r, j = 0,1, ... , m
- 1.
We denote the extended path on [0, +00) still by 'Y. Fix a ~ E 1-l(2n) and a real number Q E (1,2). For any (r,x) E and mEN, we define its m-th iteration xm : R/(mrZ) -+ R 2 n by 'Vjr
~
t
~
(j
+ l)r,
j = 0,1,2, . .. , m - 1.
(4.5)
J(~,Q)
(4.6)
We still denote by x its extension to [0, +00). Definition 4.2 For any 'Y E p,.(2n), we define
'Vm E N.
(4.7)
The mean index ib, m) per mr for mEN is defined by
~ ( 'Y, m )
Z
=
I'
1m k-++oo
ib, mk) . k
(4.8)
For any M E Sp(2n) and wE U, we define the splitting numbers str(w) of M at w by str(w) = lim iw exp(±.;=T<)b) - iwb), (4 .9) <-+0+
for any path 'Y E P,.(2n) satisfying 'Y(r) = M . For ~ E 1-l(2n) and Q E (1,2), let (r,x) E J(~,Q). We define S+(x)
s:;'(,.) (1),
(4.10)
(i(x,m),v(x,m))
=
(ibx,m),vbx,m)),
(4.11)
i(x,m)
=
ibx,m),
(4.12)
for all mEN, where 'Yx is the associated symplectic path of (r, x).
Dynamics on compact convex hypersurfaces in R 2n
279
As proved in [29], the mean index i(-y, m) is always a finite real number, and the splitting numbers topologically defined above are independent of the choice of f. A complete algebraic characterization of splitting numbers is given by Theorem 4.11 of [29]. Note that the above Maslov-type index theory (i1 (-Y), III (-y)) for, E PT (2n) was first defined by C. Conley and E. Zehnder in [5] of 1984 when n ~ 2 and , E P T (2n) is I-nondegenerate, by Y. Long and E. Zehnder in [33] of 1990 when n = 1 and, E P T (2n) is I-nondegene~~te, by Y. Long in [23] and C. Viterbo in [40] in 1990 independently when, EP T (2n) is I-degenerate and is the fundamental solution of some linear Hamiltonian system with continuous symmetric T-periodic coefficients, and by Y. Long in [26] of 1997 for any, E P T (2n) being I-degenerate. The index function (iw(-y),lIw(-y)) with W E U, the Maslov-type mean index i (-y,m), and the splitting numbers st(w) were defined by Y. Long in [29] of 1999.
4.2
Variational setting of the problem
Fix a ~ E 1i(2n) and an Q E (1,2). To solve the given energy problem (1.4) as in [10] but with our .J in (1.1) instead, we consider the following fixed period problem with He>. defined by (1.3).
i(t) { z(I)
=
=
JH~(z (t)) ,
"it E R,
z(O).
Define
E = {u E L(e>.-l)!e>.(R/Z, R2n) I
10
(4.13)
1
udt = O}.
(4.14)
The corresponding Clarke-Ekeland dual action functional f : E --+ R is defined by 1 1 (4.15) f(u) = {-(Ju, TIu) + H~( -Jundt, o 2 and f E C 2(E,R). Here TIu is defined by ftTIu = u and fo1 TIudt = 0, and the
1
usual dual function H; of He>. is defined by H~(x) =
sup ((x , y) - He>. (y)) .
(4.16)
yER2n
Here (' , .) denotes the standard inner product of R 2n. Following §V.3 of [10], we denote by "ind" the Fadell-Rabinowitz Sl-action cohomology index theory for Sl-invariant subsets of E defined in [10] (cf. also [14] of E. Fadell and P. Rabinowitz for the original definition) . For [f]e == {u E Elf (u) ~ c}, the following critical values of f are defined ck=inf{c
"ikEN.
(4 .17)
Yiming Long
280
Using the Ma.slov-type index theory defined above, the results of Ekeland et al contained in (V.3.21) , (V.3.22), Proposition V.3.3, and Theorem V.3.4 in the Section V.3 of [10] can be rephrased as follows . -00
< Cl
# j("£)
= =
infEf(u):s
C2
uE
+00,
if
Ck
=
:s ... :s Ck :s Ck+l :s ... < 0,
(4.18)
for some kEN.
(4.19)
Ck+l
For any given kEN, there exists (r,x) E .7(1: , a) and mEN such that for u~(t) = (mr) (a- l)/ (2-a):i; (mrt) ,
O:S t
:s 1,
(4.20)
there hold f'(u~)
i(x,m)
0,
<
f(u~)
= Ck,
(4 .21)
2k-2+n:Si(x,m)+v(x,m)-1.
(4 .22)
Based upon (4.21) and (4.22), the following definitions are introduced in [27] . Definition 4.3 For any 1: E 1l(2n) and a E (1, 2), (r, x) E .7(1:, a) is (m, k)variationally visible , if there exist some m and kEN such that (4.21) and {4 .22} hold for u~ defined by {4.23}. We call (r, x) E .7(1:,a) infinite variationally visible, if there exist infinitely many (m,k) such that (r,x) is (m , k)-variationally visible. We denote by V(1: , a) (or Voo (1: , a)} the subset of j(1:) in which a representative (r , x) E .7(1:, a) of each [(r, x)] is variationally visible {or infinite variationally visible} .
4.3
Main new ideas and sketch of proofs
We explain the ideas in the proof of Theorem 3.1 first. As in [27], we define the m -th index interval of (r , x) E .7(1:, a) by the closed interval Im(r , x) = [i(x , m) ,i(x, m)
We call the set I(r ,x) =
+ v(x , m)
-1],
U Im(r, x),
(4.23)
(4.24)
m 2': l
the index cover set of (r,x) . Now (4.22) can be rephrased as
2N - 2 +n C
U [(1", x)]E j(E)a)
I(r , x) .
(4.25)
Dynamics on compact convex hypersurfaces in R 2n
281
We call integers in the sequence 2N - 2 + n effective integers. From our observations on the weakly non-resonant ellipsoid as well as the study on the case of 1£(4) in [31], we noticed that in order to maximize the effect of the Fadell-Rabinowitz Sl-index theory, in stead of the index interval Lm(T,X), we should consider the largest open interval which contains Lm(T,X), possesses no part of any other index interval of (1', x). and still relates to the Fadell-Rabinowitz index suitably. This leads to our introduction of the index jump of (T,X). Definition 4.4 ([35J) For ~ E 1£(2n) and 0: E (1,2), we define the m-th index jump 9m(T, x) of (1', x) E J(~, 0:) by the open interval
9m(T, x) = (i(x, m) For
~ E
1£-(2n), we have
+ v(x, m)
Voo(~,o:)
- 1, i(x, m
+ 2)).
(4.26)
i: 0 and write (4.27)
We observe that there are infinitely many chances that the index jumps of all the q closed characteristics contain common intervals, i.e. there exist infinitely many (N,m1, ... ,m q) E Nq+1 such that
n q
[2N - "-I, 2N + "-2]
c
92mj-1 (Tj, Xj),
(4.28)
j=l
where "-I
"-2
=
"-I (~,
0:)
= "-2(~' 0:) =
min (i(xj, 1) + 2S+(x) - v(Xj, 1)),
(4.29)
m~n
(4.30)
l~J~q l~J~q
(i(xj, 1) - 1).
By the Fadell-Rabinowitz Sl-index theory, there is a one to one correspondence between the effective integers contained in the left hand side interval of (4.28) and the index jumps on the right hand side of it. Together with comparisons on "-I, "-2, and l?n(~), we have q
> #((2N-2+n)n[2N-"-1,2N+"-2]) > l?n(~)'
(4.31)
Therefore the existence and size of the interval [2N - "-1, 2N + "-2] in (4.28) is very crucial for our multiplicity results. The proof of (4.28) depends on the following new abstract precise iteration formula and the iteration inequality of the Maslov-type index theory;
Yiming Long
282
Theorem 4.1 ([35J) For any (r,x) E J(~,a) there holds i(x,m)
= m (ib, 1) + St(l) -
C(M))
L
+2
E(;:)SM(eV-1S)
SE(O,27r)
-(St(l)
+ C(M)),
where M = 1'",(r), E(a) "L..O
= min{k
(4.32)
"1m E N,
E Z I k ~ a} for all a E R, and C(M)
=
Theorem 4.2 ([35]) For any (r,x) E J(~,a) there holds i (x, m
.( x, ) m - v (x, m )
+ 1)
Z
>
i(x, 1) - n
+ 1,
.( x,l ) - eb",(r)) 2 +1
~ Z
"1m EN.
(4.33)
Note that the proof of Theorem 4.1 depends on the Bott-type iteration Formula of the Maslov-type index theory and the complete understanding of the splitting numbers established in [29]. The proof of Theorem 4.2 depends on the precise iteration formulae of the Maslov-type index theory established in [31]. This proof in fact gives a universal method to detect and prove any iteration inequalities of the Maslov-type index theory. Theorem 4.2 guarantees the index intervals and index jumps lining up nicely. By Theorem 4.1, the change of i(x, m) in m consists of a linearly increasing term m(ib, 1) + St(l) - C(M)), rotator terms E(,;:) with SM(eV-1S) > 0, and a bounded term. Then the control of the location and the size of the index jumps 92m;-1(rj,Xj) for 1 ~ j ~ q mainly depends on the control of all the rotators in terms of the iteration time 2mj - l's for 1 ~ j ~ q so that they get the largest jump simultaneously. Then the problem is reduced to the study of dynamics on tori . By our study on closed additive subgroups of the standard tori, the proof of Theorem 3.1 is completed. The proofs of Theorems 3.2 and 3.3 follow from the complete algebraic understanding of the splitting numbers given by Theorem 4.11 of [29]. Given a hypersurface ~ E 1is(2n), the solution orbits of (1.1) are devided into two sets, symmetric ones and non-symmetric ones. A symmetric orbit has Maslov-type index increasing faster than the usual ones, which allows more effective integers corresponding to such orbits located in the interval on the left hand side of (4.28) . On the other hand, non-symmetric orbits always appears as pairs {[(r, x)], [(r, -x)]), and both (r, x) and (r, -x) possess the same index, nullity, and splitting numbers. Then in this case one effective integer in the interval on the left hand side of (4.28) corresponds to two different orbits [(r, x)] and [(r, -x)]. In such a way, Theorem 3.4 is proved.
Dynamics on compact convex hypersurfaces in R 2n
283
Our main idea in the proof of Theorem 3.5 is to show the existence of one closed characteristic [(Tj,X j )) found by Theorem 3.1 which makes both equalities hold in (4.33) for the chosen iteration time m = 2mj. Then it must be elliptic. This closed characteristic is minimal in a certain sense. The proof of Theorem 3.6 depends on the understanding of the mean index sequence of iterations of closed characteristics. When ~ E 1i-(2n), the corresponding mean indices of iterations of closed characteristics must strictly increase suitably. Then we prove that if any two of the solution orbits found by Theorem 3.1 are rational, by our choice of the iteration times, the two corresponding mean indices must be equal to each other, and then yields a contradiction. The proof of Theorem 3.7 depends on the precise iteration formulae of the Maslov-type index theory established in [31). By these formulae , all the closed characteristics on ~ can be classified into finitely many families according to their different index iteration patterns. Then in each case we prove that (4.25) can not hold if there exist fewer than two elliptic closed characteristics. To prove Theorem 3.8, we further observe that the elliptic solution found in the Theorem 3.5 corresponds to a vertex X of some cube [0, l)k for some kEN. By the proof of Theorem 3.6, when n 2:: 2 we can find two such vertexes, and then we prove that they produce two different elliptic closed orbits.
5
Further considerations and open problems
The aim of the study on the problem (1.1) is three folds . The first is to completely understand the structure of the set of closed characteristics on any given hypersurface ~ E 1i(2n), and then to further understand the characteristic flow on ~. The second is to understand the structure of the set 1i(2n) . The third is to use it as the most typical example in the study of the closed solution orbits of the Reeb vector fields on general contact manifold. Our recent results give answers to these problems at different degrees. Based on our Theorems 3.1 to 3.8 and earlier results mentioned in the above section 2, it is natural to propose the following conjectures. Conjecture 1 There holds
{# j(~) I ~ E 1i(2n)} = {k E N I [~) + 1 ~ k ~ n} U {+oo} .
(5 .1)
This conjecture actually includes three steps. The first step is to construct (or prove the existence of) a particular hypersurface ~ E 1i(2n) such that # j(~) = [~) + 1. The second step is a generalization of the result (2.8) of Hofer-Wysocki-Zehnder to higher dimensions , i.e. # j(~)
2:: n + 1 implies
# j(~) =
+00,
for
~
E 1i(2n).
(5.2)
Yiming Long
284
The third step is to study how the # j(,£) decreases from n to [n/2] + 1 when '£, for example the weakly non-resonant ellipsoid, loses its symmetry. Conjecture 2. For any '£ E 1l-(2n) , all the closed characteristics on '£ are elliptic. Here we would also like to remind the readers on the following reduced version of the conjecture (2.10) based on our Theorem 3.5: Conjecture 3. # je('£) 2:: 1 for any '£ E 1l00(2n). More generally, it is interesting to know the global structure of the set j(,£) for any '£ E 1l 00 (2n). It is a interesting problem to clarify the relation between the contact (or corresponding symplectic) invariants of the given contact manifold and the number of periodic orbits of the Reeb vector field on it. Such a relation should include the case of hypersurfaces in 1l(2n), specially our Theorems 3.1-3.8. It is interesting to know the relation between our (In('£) and the possible topological invariants. For this problem, we refer the readers to the recent result [22] for an exciting work. As pointed out in [31], the study of the stability of closed characteristics can be reduced to the following conjectures on primes in the number theory. Given any pEN, we define an integer pair set Y(p) = {(pn - 1,pn + 1) In EN}.
(5.3)
Conjecture 4. For any pEN. the set Y(p) contains infinitely many prime number pairs. Note that this is a slight general version of the conjecture on twin primes. Given any cp E [0,1] \ Q, we define an integer set Z(cp)
=N
\ {2n+ 2[ncp]
+ lin EN}.
(5.4)
Conjecture 5. For any cp E [0,1] \ Q, the set Z(cp) contains infinitely many prime numbers. It is even more interesting if one can give dynamical system proofs to these conjectures on prime numbers.
References [1] A. Ambrosetti and G. Mancini, On a theorem of Ekeland and Lasry concerning the number of periodic Hamiltonian trajectories. J. Diff. Equa. 43 (1982) , 249-256. [2] T. Bartsch, A generalization of the Weinstein-Moser theorenms on periodic orbits of a Hamiltnian system near an equilibrium. Ann. Inst. H.Poincare, Anal. Non lineaire. 14, (1997) 691-718.
Dynamics on compact convex hypersurfaces in R 2 n
285
[3] H. Berestycki, J. M. Lasry, G. Mancini, and B. Ruf, Existence of multiple periodic orbits on star-shaped Hamiltonian surfaces. Comm. Pure Appl. Math. 38 (1985) 252-290. [4] R. Bott, On the iteration of closed geodesics and the Sturm intersection theory. Comm. Pure Appl. Math. 9.(1956) . 171-206. [5] C. Conley and E. Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian equations. Comm. Pure Appl. Math. 37.(1984). 207-253. [6] G. Dell'Antoio, B. D'Onofrio, and I. Ekeland, Les system hamiltoniens convexes et pairs ne sont pas ergodiques en general. C. R. Acad. Sci. Paris, t .315. Series I. (1992). 1413-1415. [7] D. Dong and Y. Long, The iteration formula of the Maslov-type index theory with applications to nonlinear Hamiltonian systems. Trans. Amer. Math. Soc. 349. (1997). 2619-2661. [8] I. Ekeland, Une theorie de Morse pour les systemes hamiltoniens convexes. Ann. Inst. H. Poincare Anal. Non Linearire. 1. (1984). 19-78. [9] I. Ekeland, An index theory for periodic solutions of convex Hamiltonian systems. In Nonlinear Funct. Anal. and its Appl. Proc. Symposia in Pure Math. 45-1. (1986) . 395-423. [10] I. Ekeland, Convexity Methods in Hamiltonian Mechanics. Springer. Berlin. (1990). [11] I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and their closed trajectories. Comm. Math. Phys. 113 (1987), 419-467. [12] I. Ekeland and J .-M. Lasry, On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface. Ann. of Math. 112 (1980), 283-319. [13] I. Ekeland and L. Lassoued, Multiplicite des trajectories formees d'un systeme hamiltonian sur une hypersurface d'energie convexe. Ann. INP Analyse non lineaire. 4 (1987), 1-29. [14] E. Fadell and P. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems. Invent. Math. 45. (1978) 139-174. [15] M. Girardi, Multiple orbits for Hamiltonian systems on starshaped energy surfaces with symmetry. Ann. IHP Analyse non lineaire. 1 (1984) 285294.
286
Yiming Long
[16] J. Han and Y. Long, Normal forms of symplectic matrices (II) . Nankai Inst. of Math. Nankai Univ. Preprint. (1997). Acta Sci. Natur. Univ. Nankaiensis. 32 (1999) 30-41. [17] H. Hofer, A new proof of a result of Ekeland and Lasry concerning the number of periodic Hamiltonian trajectories on a prescribed energy surfaces. Boll. UMI. 6 (1982) 931-942. [18] H. Hofer, K. Wysocki, and E . Zehnder, The dynamics on three-dimensional strictly convex energy surfaces. Ann. of Math . 148 (1998) 197-289. [19] V. J . Horn, Beitriige zur Theorie der kleinen Schwingungen. Zeit. Math. Phys . 48 (1903) 400-434. [20] A. Liapunov, ProbIeme general de la stabilite du mouvement. Russian edition (1892), Ann. Fac. Sci. Toulouse 9 (1907) 203-474. [21] C. Liu, Y. Long, and C. Zhu, Multiplicity of closed characteristics on symmetric convex hypersurfaces in R2n. Nankai Inst. Math. Preprint No. 1999-M-003. Submitted. [22] G. Liu and G. Tian, Weinstein conjecture and GW invariants. Preprint. (1997) [23] Y. Long, Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems. Science in China (Scientia Sinica) . Series A. 7. (1990) . 673-682 . (Chinese edition), 33. (1990) . 1409-1419. (English edition). [24] Y. Long, The structure of the singular symplectic matrix set. Science in China (Scientia Sinica) . Series A. 5. (1991) . 457-465. (Chinese edition), 34. (1991). 897-907. (English edition). [25] Y. Long, The Index Theory of Hamiltonian Systems with Applications. (In Chinese). Science Press. Beijing. (1993) . [26] Y. Long, A Maslov-type index theory for symplectic paths. Top. Meth. Nonl. Anal. 10 (1997) 47-78. [27] Y. Long, Hyperbolic closed characteristics on compact convex smooth hypersurfaces in R2n . Nankai Inst. of Math. Nankai Univ. Preprint. (1996) . J. Diff. Equa. 150 (1998) , 227-249. [28] Y. Long, The topological structures of w-subsets of symplectic groups. Nankai Inst. of Math. Nankai Univ. Preprint. (1995). Acta Math. Sinica. English Series. 15. (1999) 255-268.
Dynamics on compact convex hypersurfaces in R 2n
287
[29] Y. Long, Bott formula of the Maslov-type index theory. Nankai Inst. of Math. Nankai Univ. Preprint. (1995, Revised 1996, 1997) . Pacific J. Math . 187 (1999), 113-149. [30] Y. Long, The Maslov-type index and its iteration theory with applications to Hamiltonian systems. Third School on Nonlinear Analysis and Applications to Differential Equations. (10. 12-30,1998). ICTP Lecture Notes. SMR 1071/2 . . Proc. of Inst. of Math . Academia Sinica. to appear. [31] Y. Long, Precise iteration formulae of the Maslov-type index theory and ellipticity of closed characteristics. ICTP Preprint (1998). Nankai Inst. Math. Preprint No. 1999-M-001. Advances in Math. to appear. [32] Y. Long and D. Dong, Normal forms of symplectic matrices . Nankai Inst. of Math. Nankai Univ. Preprint . (1995) . Acta Math . Sinica. to appear. [33] Y. Long and E. Zehnder, Morse theory for forced oscillations of asymptotically linear Hamiltonian systems. In Stoc. Proc. Phys. and Geom. S. Albeverio et al. ed. World Sci. (1990) . 528-563. [34] Y. Long and C. Zhu, Maslov-type index theory for symplectic paths and spectral flow (II). Nankai Inst. of Math. Nankai Univ. Preprint. (1997) . Revised 1998. Chinese Ann. of Math. 21B :1(2000), 89-108. [35] Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in R2n . Nankai Inst. Math . Preprint No. 1999-M-002. Submitted. [36] J . K. Moser, Periodic orbits near an equilibrium and a theorem by A. Weinstein. Comm. Pure Appl. Math. 29 (1976), 727-747. [37] P. Rabinowitz, Periodic solutions of Hamiltonian systems. Comm. Pure Appl. Math. 31. (1978) . 157-184. [38] P. Rabinowitz, Minimax methods in critical point theory with applications to differential equations. CBMS Regional Conf. Ser. in Math . no.65. Amer. Math. Soc. (1986). [39] A. Szulkin, Morse theory and existence of periodic solutions of convex Hamiltonian systems. Bull. Soc. Math. France. 116 (1988) , 171-197. [40) C. Viterbo, A new obstruction to embedding Lagrangian tori. Invent. Math. 100. (1990) 301-320. [41] A. Weinstein, Normal modes for nonlinear Hamiltonian systems. Inven . Math. 20. (1973).47-57.
288
Yiming Long
[42] A. Weinstein, Periodic orbits for convex Hamiltonian systems. Ann. of Math. 108. (1978). 507-518. [43] V. Yakubovich and V. Starzhinskii, Linear Differential Equations with Periodic Coefficients. New York, John Wiley & Sons. (1975). [44] C. Zhu and Y. Long, Maslov-type index theory for symplectic paths and spectral flow (l). Nankai Inst. of Math. Nankai Univ. Preprint. (1997) . Revised 1998. Chinese Ann. of Math. 20 B. (1999) 413-424.
Gromov-Witten invariants on compact symplectic manifolds with contact type boundaries and applications Guangcun LuI f~anKaI
!nStltute ot MatnematIcs, fllanKaI umverslty Tianjin 300071, P. R. China (E-mail: [email protected])
Abstract In this note we define the Gromov-Witten invariants on any compact symplectic manifolds with contact type boundaries and outline an application of it to the topological rigidity of Hamiltonian loops on this class of manifolds.
In the past years the Gromov-Witten invariant theory on the closed symplectic manifolds was established and studied well(cf.[Gr][Wl][W2][Rl][RT][KM] [McSal][K][FO][LT2][R2][Sie]). Clearly, it is natural and necessary to establish this theory on the symplectic manifolds which are noncompact or with boundaries in both the theory and the application. In [Lu] author defined the Gromov-Witten invariants on a class of noncom pact symplectic manifolds. As a special case the Gromov-Witten invariants on any weakly monotone compact symplectic manifolds with contact type boundaries were given. In addition we also studied topological rigidity of Hamiltonian loops with compact support that is a generalization of results in [LMP] . In this note we shall define the Gromov-Witten invariants on any compact symplectic manifolds with contact type boundaries and outline the general versions of Corollary 6.3 and 6.17 in [Lu].
1
J -convexity and hypersurfaces of contact type
In this section we will review some special important notions. Let (M, J) be an almost complex manifold of dimension 2n and SCM a smooth compact 1 Partially
supported by the NNSF of China.
289
Guangcun Lu
290
e
connected oriented hypersurface. J determines a complex vector bundle := T S n J(T S) over S of rank n - 1, which is a real vector subbundle of T S. For a xES we take a base Ul," ' , Un, JxUl,"', Jxu n for {~, and v E TxS such that
form a base for TxS and give the specified orientation of S . We call the part directed by Jxv the "interior' of S and one directed by -Jxv the "exterior' of S . It is easily proved that there exists a smooth nowhere vanishing I-form a on S such that a(v)
> 0 and
{~
= Ker(a x ).
Such a I-form is called the defining I-form of the sub bundle e. Moreover, one can prove that the defining I-form of S is unique up to the multiplication by a positive smooth function factor . Definition 1.1 Let a be a defining I-form of S, if da(v, Jv) > 0 for every nonzero vector vEe, S is said to be J -convex. Similarly, S is called J -concave ifda(v,Jv)
< 0 for all nonzero vectors vEe.
It is easily proved that this definition is independent of the choice of the
defining I-forms. Notice that there always exist an open neighborhood U of S and a smooth function p : U -+ lR such that 0 E lR is an regular value of it, S = p-l(O) and {x E U I p(x) case one can easily show that
< O} is contained in the interior of S. In this
(1.1)
a := -(J*dp)ls
e.
is a defining I-form of By the definition it is not difficult to prove that S is also J'-convex with respect to any almost complex structure J' on M such
e e'
that = and Jle = J'le" The following property is the most important one for J-convex surface. Lemma 1.2([E][Mc1]) Let S be a J-convex hypersuface in an almost comThen any J-
plex manifold (M, w) and W the union of it and its interior.
holomorphic curve contained in W can not touch S at an interior regular point. Proof Let ~ be a compact Riemann surface and U : ~ -+ M a nonconstant J-holomorphic curve with u(~) C W . We want to prove U({ z E Int(~) I du(z) =F O}) n S
= 0.
Take an open neighborhood U of S and a smooth function p : U -+ lR as above. Then (1.1) gives a defining I-form of Choose a € > 0 so small that [-€,€] only contains the regular values of p and p-l ([ - €, €]) C U.
e.
291
Gromov-Witten invariants on compact symplectic manifolds
Assume that there exists Zo E Int(~), du(zo) i:- 0 such that u(zo) E S. Since the critical points of J-holomorphic curves are isolated, by choosing a local coordinate chart we may assume that there exists a J-holomorphic disk v : D = {I z I ~ I} -+ M such that (1.2)
v(O) =
XES,
i:- 0 Vz ED.
v(D) C p-l ([ - f , 0]), dv(z)
Writting z = s + it, since I := po v : D -+ ~ attains its maximum at z = 0 we get dp(x)(8s l(0» = dp(x)(8d(0» = O. and thus
8 sl(0), 8d(0) E TxS \ {O}.
(1.3)
But 8 s 1 + J(v)8d = 0 we obtain that both 8 s l(0) and 8d(0) belong to ~;. Denote by (3 := -J*dp. This is a smooth I-form on an open neighborhood U of Sand (3ls = a. From the J-convexity of S it follows that
d(3(v(0»(8s l(0), J(v(0»8 s l(0» = da(v(0»)(8 s l(0), J(v(0»8 s l(0» > O. Thus for sufficiently small 15 > 0 it holds that
d(3(v(z»)(8sJ(z) , J(v(z»8 s l(z» > 0, Vz E Dli := {Izl ~ 15}.
(1.4)
Let dC
= Jod,
where Jo denotes the standard complex structure on C. By the
definition
dd C 1= -6.(I)ds 1\ dt on Dli·
(1.5)
o
Since dC I = JodI = Jodp 0 dv = J 0 v*(dp) = J (J 0 v*)*dp = v· 0 J*dp = -v*(3 we arrive at
dd C 1= d(v*
0
o
0
(v*)*dp = (v*
0
Jo)*dp
=
J*dp) = -v*d(3 = -d(3(8s v, 8t v)ds 1\ dt.
Hence this and (1.4) together lead to (1.6)
6.(1) = d(3(8s v,8t v) = d(3(8s v,J(v)8s v) > 0 on Dli ·
But 118 D 6 ~ O. The Maximum principle leads to p(v(O» < O. This contradicts to x = v(O) E S.
Ilrnt(D6)
< O. Specially, 0
Guangcun Lu
292
Remark 1.3 In the past arguments, for example, Lemma 1.4 in [Mel), they, in fact, only considers the case that 0 is a regular point, and neglects that of 0 being a critical point. From the above proof we may know that in the case of
8s v(O) = 8t v(O) = 0 it is impossible to use this method to get the conclusion that x = v(O) ~ S. However, later we will show that there exists another almost = and such that not only S is complex structure J' near S for which J'-convex but also every J'-holomorphic curve contained in W cannot touch S from the inside.
e e'
A class of important pseudo-convex hypersurface comes from the hypersurface of contact type([We)). Recall that a compact smooth hypersurface S in a symplectic manifold (M,w) is said to have contact type if there exists a smooth vector field X in a neighborhood U of S that is transverse to S and Lxw = do ixw
+ ix
0
dw = w. An equivalient statement is that there exists
a contact form 0 on S such that do = wis. Define the part directed by X the outside of S. It is easily checked that this definition is independent of the choice of such X and o. Thus different choices of X may give the same orientation on S by ixw n . Later, the orientation on S always means this one without special statements. The following lemma gives the relation between J-convex hypersurface and one of contact type. Lemma 1.4 For every hypersurface S of contact type in a symplectic manifold (M,w) there exists a w-compatible almost complex structure J E .J(M,w) such that S is not only J -convex with respect to the orientation just defined but also has the property:
(1.7) every J-holomorphic curve in the interior of S cannot touch it from the inside. Conversely, if an oriented hypersuface S C (M, J) is a J -convex there always exists a contact form on S. Proof The second claim is clear. In fact, let p : U -+ IR give a defining I-form
e
e
-(J*dp)ls of := TSn (JTS) as above. Then = Kera is a sub bundle of T S of rank 2n - 2, and do( v, Jv) > 0 for any nonzero vector vEe because Hence of J -convexity of S . The latter implies that do is nondegenerate on o is a contact form on Sand o2n-1 gives the specified orientation of S.
0=
e.
For the first claim Proposition 8.14 in [ABKLR) showed that there exists a E .J(M,w) such that S is J-convex. The remainder is to find such a J
Gromov-Witten invariants on compact symplectic manifolds
293
also satisfying (1.7). To this goal let us consider the symplectizatins of the contact manifold (S,a), (N,w) := (IR x S,d(eta» . The usual implict function theorem arguments show that there exist a € > and a diffeomorphism from (-2€,2€) x S onto an open neighborhood U C p-1([-2€,2€)) of Sin M,
°
(1.8)
~: (-2€, 2€)
x S -t U, ~(O,x)
= x,
"Ix E S,
that maps (-2€,O) x S into the inside of S( in fact, p-1([-2€,0))). Denote by XO/ the Reeb vector field determined by ixaa = 1 and iXada = 0. Then one gets a natural splitting TS = IRXO/ EB Let 7r0/ : TS -t be the projection along IRXO/ given by v t-+ v - a(v)XO/. These may pick out a class of important almost complex structures in .J(N,O) as follows:
e.
(1.9)
J(t, x)(h, k)
e
= (-a(x)(k), J(x) (7rO/k) + hXO/(x»,
where hEIR ~ TslR, k E TxS and .i E .J(e, dale). The corresponding compatible Riemann metric is given by (1.10)
9J(t, x)((h 1, k 1), (h 2, k 2» = 0
0
(id x J)
=9j(7rO/k 1, 7rO/k2 ) + h1h2 + a(x)(kda(x)(k2) ' J -convex and its inside is contained in (-00,0) x S. The important feature of this class of almost complex structures is Fact 1.5([H)). Given a J-holomorphic disk u : D = {Izl ::; I} -t N contained in (-00,0) x S, if u(8D) C (-00,0) x S then u(D) C (-00,0) x S, i.e. u cannot touch S = {O} x S from the inside. Indeed, write u = (a, u) with a E IR and u : D -t S . From the proof of Lemma 19 in [H) we have It is easily checked that every hypersuface St = {t} x S is
~a = ~[J7r0/8sul~J + 17r0/8tul~J) 2:
°
on D .
But alaD < 0. Hence the strong maximum principle leads to the conclusion. Now using the diffeomorphism ~ in (1.8) we may define an almost complex structure on the open neighborhood ~(( -2€, 2€) X S) of S as follows: (1.11)
J w :=
d~ 0
J-
0
(d~)
-1
.
Guangcun Lu
294
It is clear that S is J<J?-convex.
But it is more important that every J<J?-
holomorphic curve contained in the interior of S cannot touch S from the inside. Notice that J<J? E .J(<J>(( -210,210) x S), w) . Denote by g<J? := wo (id x J<J?) the Riemann metric on <J>(( -210,210) X S) . Using the technique of a partition of unity it can be extended to a Riemann metric g on M with gl<J>([-f, 10] x S) = g<J? Then the standard technique gives rise to J E .J(M,w) satisfying: JI<J>([-f,f] X S) =
o
J<J? Clearly, this J satisfies the requirements.
2
Gromov-Witten invariants on compact symplectic manifolds with contact type bound. arIes
In [Lu] we defined the Gromov-Witten invariants on the compact semipositive symplectic manifolds with contact type boundaries as a special case of those on
a class of noncompact symplectic manifolds. In this section we will define them in general case. Let (M, w) be a compact symplectic manifold with smooth boundaries. Then the boundary 8M has a natural orientation defined by the volume form iv(wn)laM for an outward pointing vector field v along 8M . A contact form ,X on 8M is called positive if ,X 1\ (d,X)n-l = fiv(wn)laM for some smooth positive function f on 8M . Following A.Weinstein([WeJ) we call (M,w) to have contact type boundary if there exists a positive contact form ,X on 8M such that d,X = WlaM ' For such contact form ,X there always exist 10 > symplectic embedding of co dimension zero
°
and a
'ljJ: ((-10, 0] x 8M,d(e t ,X)) --t (M,w)
such that 'ljJ(O,x) = x for any x E 8M. Using it we get another compact symplectic manifold with contact type boundaries (M, w) as follows :
M
=M
U aM x{O}
8M x [0, 2]'
_ { w o n Mj w = d(et,X) on 8M x [0,2] .
Here t is the second coordinate. Let J), E .J(M,w) be the almost complex structure as constructed in Lemma 1.4. Then it it clear from construction of it that this J), may be extended onto all of M. We denote it by i)'. Notice that
Gromov-Witten invariants on compact symplectic manifolds
295
every hypersurface contact type {t} x 8M has the same property as 8M == {O} x 8M for any t E (-f, 2]). From now on we omit the subscript A in J>. without occuring of confusions. For any 0 < 8 < f we denote by (2.1) .J(M, 8M,w; 'I/J, J,8) := {J' E .J(M,w) I J'11/t((-o,Ojx8M) = JI1/t((-o,Ojx8M)}, { .J(M,8M,wj J) := {J' E .J(M,w) equals J near 8M}. Both are contractible nonempty subset of .J(M,w), and for any given JI E .J(M, 8M, Wj J) there exists a 8 > 0 such that JI E .J(M, 8M, Wj 'I/J, J,8). By perturbing J in .J(M, 8M, w; 'I/J, J,8) we may assume that .T is regular and fix it in the following arguments without special statements. Moreover, we are only satisfied pointing out how the arguments in [LT2] are realized in the present case. Let (1:: = U~l 1:: i j Xl, ... ,Xk) be a k-pointed semistable curve of genus g( which, by the definition, is connected[FO]), and 1I"i : ~i -t Ei denote the normalization of Ei . A continuous map u : E -t M is called J-holomorphic if every lift u 0 1I"i : ~i -t M is J-holomorphic. It is said to be stable if Ei contains at least three special points( double or marked points) when UiI::i is constant. Denote by [u, Ej Xl, ... ,Xk] the isomorphism class of the stable Jholomorphic map (u, 1::j Xl,"', Xk). Applying Lemma 1.4 to every component u 0 1I"i we"get u(E) n 'I/J(( -8,0] x 8M) = 0. This is a key for our arguments. Define u.([1::]) := Li(U01l"i).([~i]) the homology class represented by u. Given a A in the free part of H 2 (Mj Z) we denote by Mg ,k(M,w, J, A) the space of all isomorphism classes of k-pointed stable J-holomorphic curve of genus g. This is compact. Let Mg,k be the moduli space of Riemann surfaces of genus 9 with k marked points, and Mg,k be the Deligne-Mumford compactification of Mg,k consisting of the isomorphism class of all genus 9 stable curves with k marked points. Denote by --
EV: Mg,k(M,w,J,A) -t M the evaluation map given by EV([u, Ej Xl, '
"",
k
Xk]) = (u(xd,""' U(Xk)), and
the map given by successively contracting the unstable component of the domain of stable maps. Then, roughly speaking, the Gromov-Witten invariants
296
Guangcun Lu
are the homology class represented by the image of 7rg ,k x EV. To define them one need to embed Mg,k (M, w, J, A) into a larger space so as to get a "virtual fundamental class" of it. To this goal the notion of the stable CI-stable map( l ~ 1) with k marked points and of genus 9 was introduced in [LT2]. Let
-rA (M, g, k) be the space of all isomophism classes of such a map representing A. Then it contains Mg ,k(M,w,J,A) as a subset, and EV, 7rg,k are still well-defined on it. Notice that unlike the case of M being closed manifold the present -rA (M, g, k) has nonempty boundary. It is the property of the image of every map in Mg ,k(M,w , J,A) not intersecting with "p((-6, O] x 8M) that the space M 9 ,k (M, w, J, A) is contained in the interior of -rA (M, g, k) . That is, it does not intersect with the boundary part of it. Hence, by the compactness of Mg,k (M, W, g, k) we may choose an open neighborhood of it which is not intersecting with the boundary part of-rA (M, g, k) . So one only needs to repeat the arguments in [LT2] to define a generalized bundle E over -rA (M, g, k) and a natural section J given by the Cauchy-Riemann operator. They together give rise to a generalized Fredholm orbifold bundle with the natural orientation and of index r := 2c1(M)(A)+2k+(2n-6)(1-g). When l ~ 2 Theorem 1.2 in [LT2] determines an Euler class e([J : -rA(M,g,k) M ED in Hr(~(M,g,k);Q). But the inclusions FA(M,g,k) := F:(M,g,k) '-+ -rA(M,g,k) are all homotopically equivalient. Thus they give rise to a unique Euler class e([J : FA(M,g,k) homomorphism
M
ED
in Hr(FA(M,g , k);Q), which in turn defines a
by taking slant product of this Euler class by cohomological class in H*(Mg,k; Q). Clearly, it satisfies
P~',~,k(O U (3) = P~',~, k(O)/7r;,k(3 for any 0,(3 E H*(Mg ,k;Q) . Define
I)i('A~9,k) : H* (M; Q)k x H* (Mg,k; Q) -+ Q by 1)i('/9,k) ((3; 01, . .. ,Ok) = EV* (7ri 01 1\ ... 1\ 7rkOk)(P~:~,k ((3)). Notice that for any two almost complex structures J1 and J 2 in .:J(M,8M,w; J) there
Gromov-·W itten invariants on compact symplectic manifolds
297
exists a 8> 0 such that they belong to J(M, 8M, w; '1/;, J,8) and therefore are homotopic in J(M,8M,w;'I/;,J,8). We get that e([<JlJI : FA(M,g,k) I-t ED = e([<Jl h : FA(M,g,k) I-t ED and thus w,J 1
(2.2)
_
w,J2
PA,g,k - PA,g,k
and
"IJIw,h _ "IJI W,J2 (A,g ,k) - (A,g ,k) ·
These show that "IJI(A~~,k) is independent of the choice of J' E J(M, 8M, w; J) . Next, for the extended almost complex structure j = j;. on M above every .1' E J(M,w) may be extended into an element ], in J(M,8M,w; j;.)( but need not to be unique). Using the natural isomorphisms i* : H(M; Z) --t H*(Mj Z) and i* : H*(M; Z) --t H*(M; Z) induced by the inclusion i : M '-+ M we define
(13· ) .- ,T.-;;;J>. (13·, ell, - ... , elk - ) , ,k) , ell, ... , elk .- ~ (A,g,k)
,T.W,;' ~(A 9
(2.3)
for 13 E H*(Mg ,k; Q) and ell E H*(M; Q), l = 1, ···, k. Here A = i*(A) and III = (i*)-l(eld for any l. Clearly, this is well-defined. However, it may depend on the contact form A above. To show that such case cannot happen we denote by Cont+ (M, 8M; w)
(2.4)
the set of all positive contact forms A on 8M such that dA = WlaM. Lemma 2.1 The space Cont(M,8M;w) is a convex set. Proof For any two AO, Al E Cont+(M, 8Mj w), by definition it holds that dAO
=
WlaM = dAI, AO
1\
(dAo)n-1
= f(illwn)laM
and
Al
1\
(dAd n- 1 = g(illwn)laM
for some positive smooth functions f and 9 on 8M, Assume that (I-to)Ao(xo)+ toAI (xo)
= 0 for some to E (0,1) and Xo
E 8M. Then
But (1- to)f + tog> 0 on 8M. Hence the right side of (2.5) is not equal to zero at any point. This contradicts to the fact that (1 - to),o(xo) + toAI (xo) = o. Thus for any t E [0,1], I-form (1 - t)AO + tAl is nowhere zero. It is easily checked that they all belong to Cont+(M,8M;w).
0
Guangcun Lu
298
Define As := (1 - 8)AO + 8Al for 8 E [0,1]. Let ~s := KerA s and Ys be the Reeb vector field of A.•. A given Riemann metric g on 8M induces an Riemann metric gs on ~s. With gs and wle. the standard arguments produces a smooth family of almost complex structures is on ~s. On the other hand, Gray stable theorem gives rise to a family of diffeomorphisms Fs : 8M -+ 8M such that
F; As = fsAo for a family of smooth nowhere vanishing functions fs on 8M. Take a smooth family of vector fields X. om M such that
Define 0 and a smooth family of embedding of co dimension zero 'l/Js : (-€, 0] x 8M -+ M such that
As above we get a family of diffeomorphic compact symplectic manifolds with contact type boundaries (Ms,w s ) as follows:
U
Ms = M
8M x [0,2]'
",.(aMx{O})~aMx{O}
_ { w o n M; Ws = d(e t As) on 8M x [0,2].
Here t is the second coordinate. Using the gluing technique we get a smooth family of almost complex stuctures is E .J (Ms, ws ) such that
for all 8 E [0,1]. Notice that there exists a family of natural symplectic diffeomorphisms
IJI s : Mo
-+ Ms, y
t-+ {
~s(t, x) (t, x)
Thus
8
t-+
IJI;J.
as M \ 'l/Jo((-€,O] x 8M); as y = 'l/Jo(t,x); as (t,x) E (0,2] x 8M.
gives rise to a smooth path in .J(Mo,8Mo,wo; io). Hence
Gromov- Witten invariants on compact symplectic manifolds
299
These show that \If('l9,k) defined by (2.3) is independent of the choice of contact form .x E Cont+ (M, 0; w), and thus is invariant under symplectomorphisms of (M,w) . We call (2.6)
\lfW
= \lfW,A
(A,g,k) -
(A,g ,k)
the Gromov- Witten invariants of a symplectic manifold with contact type boundaries. It is easily checked that they satisfy all properties that are satisfied by the usual the Gromov-Witten invariants of a closed symplectic manifold.
3
An application
As mentioned before the Gromov-Witten invariants have many important applications. Based on Seidel's work F .Lalonde, D.McDuff and L.Polterovich studied the rigidity of Hamiltonian loops([LMP]) . Precisely speaking, they proved: Theorem 3.1 If WI and W2 are two symplectic forms on a closed manifold M
of dimension 2n satisfyimg: (3.1)
A E 7r2(M), 2 - n ::;
CI (A)
< 0 => w(A) ::; 0,
then for any ¢ E 7r1 (Diff(M) , id) it holds that ¢ E Im(Hwl) n Im(8wJ if and only if ¢ E Im(HW2 ) nIm(8w1 ). Here 8 w : 7r1(Sympo(M,w)) -+ 7r1(Diff(M), id) and Hw : 7r1(HamO(M,w)) -+ 7r1(Diff(M),id) are the homomorphisms induced by the group inclusions respectively. In Corollary 6.3 and 6.17 of [Lu] we generalize this result to the case that both (M, WI) and (M, W2) are the symplectic manifold with contact type boundaries satisfying (3.1) . In a recent beautiful paper [Mc2] D.McDuff systematically studied the quantum homology of fib rations over 8 2 and obtained many deep results. As a consequence of her Theorem 1.1 it was proved that if P", -+ 8 2 is a fibration with any closed symplectic manifold (M , w) as a fibre constructed from a loop ¢ = {¢t} E Ham(M,w) then the homomorphism induced by the inclusion M '-+ P"" i* : H*(M ; JR) --t H*(P",; JR) is injective. This especially implies that Theorem 3.1 still holds even if (3.1) is removed. Using the techniques developed in §2 we can show that the same conclusion holds on any symplectic manifold with contact type boundaries. The detailed
Guangcun Lu
300
arguments will be given in the further paper. Acknowledgements This note was accomplished during author's visit to Morning Center of Mathematics. He is very grateful to Professor Gang Tian for his invitation and hospitality.
References [ABKLH) B. Aebischer, M. Borer, M, Kalin, Ch. Leuenberger and H.M.Reimann, Symplectic Geometry, Progress in Mathematics, vol. 1246, Birkhauser Verlag, 1994. [E)
Y.Eliashberg, Filling by holomorphic discs and its applications, In Geometry of low-dimensional manifolds 2, S.Donaldson and C.B. Thomes, editors, London mathematical society lecture note series 151. Cambridge university press, 1990.
[FO) K. Fukaya and K. Ono, Arnold conjecture and Gromov-Witten invariant, preprint, 1996. [Gi) B. Givental, Equivariant Gromov-Witten Invariants, preprint, 1996. [Gr) M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Inv. Math., 82 (1985), 307-347. [H] H.Hofer, Pseudoholomorphic curves in symplectisations with application to the Weinstein conjecture in dimension three, Inv . Math ., 114 (1993) , 515-563. [K)
M. Kontsevich, Enumeration of rational curves via torus actions, in Moduli space of surface, Dijkgraaf.H, Faber.C, and v.d. Geer.G, Editors. 1995, Birkhauser: Boston.p.335-368.
[KM) M. Kontsevich and Y. Manin, GW classes, Quantum cohomology and enumerative geometry, Comm.Math.Phys., 164(1994) , 525-562. [LMP) F. Lalonde, D. McDuff, 1. Polterovich, Topological rigidity of Hamiltonian loops and quantum homology, Inv. Math., 135 (1999),369-385.
Gromov- Witten invariants on compact symplectic manifolds
301
[LT1] J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Amer. Math. Soc., 11(1998), 119-174. [LT2] J. Li and G. Tian, Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds, preprint. [Lu] G. C. Lu, Gromov-Witten invariants and rigidity of Hamiltonian loops with compact support on noncompact symplectic manifolds, preprint, IHESjMj99j43. Math.DGj9905074, 1999. [Mel] D. McDuff, Symplectic manifolds with contact type boundary. Inv. Math ., 103 (1991),651-671. [Mc2] D. McDuff, Quantum homology of fibrations over S2, preprint, SG 9905092. [McSa1] D. McDuff and D. Salamon, J-holomorphic curves and quantum cohomology, University Lec. Series, vol. 6, AMS. [McSa2] D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford University Press, 1995 . [PW] T. Parker and J. Wolfson, Pseudo-holomorphic maps and bubble trees, J. Geom. Analysis., 3(1993), 63-98. [R1] Y. Ruan, Topological Sigma model and Donaldson type in Gromov theory, Math. Duke. Jour., 83, 2(1996), 461-500. [R2] Y. Ruan, Virtual neighborhood and pseudo-holomorphic curves, preprint. [R3] Y.Ruan, Quantum Cohomology and its Application, Doc.MATH.J.DMV., 1998. [RT] Y. Ruan and G. Tian, A mathematical theory of quantum cohomology, J. Diff. Geom., 43, 2(1995). [Se] P. Seidel, 7rl of symplectic automorphism groups and invertibles in quantum homology rings, Geom . Funct. Anal.,17, 6(1997). [Siej B. Siebert, Gromov-Witten invariants for general symplectic manifolds,preprint, 1996.
302
Guangcun Lu
[We] A.Weinstein, On the hypothesis of Rabinowitz's periodic orbit theorems, J. Diff. Equatins., 33(1979), 337-358.
[WI] E. Witten, Topological sigma models, Comm. Math . Phys., 118(1988). [W2] E . Witten, Two dimensional gravity and intersection theory on moduli space, Surveys in Diff. Geom. , 1(1991) , 243-310. [Ye] R. Ye, Gromov's compactness theorem for pseudo-holomorphic curves, Trans. Amer. Math. Soc., 343(1994), 671-694.
Exact Lagrangian Submanifolds In Symplectizations* Renyi Ma Department of Applied Mathmatics Tsinghua University Beijing, 100084 People's Republic of China Abstract In this article, we prove the non-existence of exact Lagrangian submanifolds in symplectization of contact manifolds which generalizes Gromov's previous work in R 2n and cotangent bundles.
§1. Introduction and results Let U be a (2n - I)-dimensional manifolds. A contact structure ~ on U is a completely nonintegrable co dimension I tangent distribution. It means that ~ can be defined, at least locally, by a I-form A with A /\ (dA)n-l f:- O. Note that if n is odd then the contact distribution ~ is automatically orientable. For an even n the existence of a contact structure implies the orientability of the ambient manifold U. In both cases, the co orient ability of ~ implies that ~ and U are both orientable. We will asume from now on that ~ is cooritable and fix its orientation. Then ~ can be globally defined by a I-form A, which is determined up to a multiplication by a positive function. Let SU = U x ]0,00[. We will still denote by A the pull back of A by the projection of SU = U x R+ on the first factor and denote by t the projection on the second. Then the form w = d(tA) defines a symplectic structure on SU(indeed, (d(tA))n = tn-1dt /\ A/\ dA n- 1 f:a). The map (x, t) ~ (x, tj f(x)) induces an isomorphism of forms d(tA) and d(tJl.) for Jl. = fA. Therefore, the symplectic manifold (SU,w) depends, up to a symplectomorphism, only on the contact manifolds (U,~) and not on the choice of the I-form A. For an n-dimensional manifolds M let us denote by ST*(M) the unit cotangent bundle of M with the contact structure ~ defined by the form ·Project 19871044 Supported by NSF
303
304
Renyi Ma
pdq. The manifold ST* M can also be considered a space of co oriented (n -1)dimensional contact elements of M. With this interpretation the plane ~x of ~ at a point x = (p, q), q EM, pET; (M), consists of infinitesimal deformations of ~x, which leaves fixed the point of contact q EM. Then the symplectization Sympl(ST*(M),~) is isomorphic to T*(M) \ M with the standard symplectic structure w= d(pdq), for more example, see[2-6]. Let (V',w(= da)) be an exact symplectic manifold and LeV' a closed submanifolds, we call L an exact Lagrangian submanifold if aiL an exact form, i.e., aiL = df. Consider an isotopy of Lagrange submanifolds in V' given by a Coo-map F' : W' -+ V' and let w' be the pull-back of the form w' to W' x [0,1]. The form w' clearly is exact, w' = di/ , where the I-form i ' on W' x t for t E [0,1]. Recall that F' is called an exact isotopy if the class [i'IW ' x t] E HI(W' = W' x t;R) is constant in t E [0,1]. Theorem 1.1 Let (SU,w(= d(tA))) be the symplectization of contact manifold (U, A), F' be an exact isotopy and let the submanifolds W' = F'(W' X t) c (SU, d(tA)), t E [0,1] be exact. Then the intersection Wo n WI is non-empty.
Theorem l.I was first proved by Gromov in [5] for closed symplectic manifolds and cotangent bundles. Gromov's argument can be extended to the symplectic manifolds with boundary of contact type, see [5,9]. Theorem 1.2 Let (SU,w(= d(tA))) be the symplectization of contact manifold (U, A), then there does not exist any exact closed Lagrangian submanifold L in (SU,w = d(tA)). For (SU,w) = (R 2n ,wo), this theorem was first proved by Gromov in [5]. Proof. The manifold (SU,d(tA)) has a canonical diffeotopy v' -+ sv' for s E [0,00). The induced isotopy on L clearly is Lagrange; it is exact if and only if the form IIIW ' is exact. The isotopied manifolds L. = s(L) are disjointed from L for s -+ 00 as well as s -+ zero, this contradicts Theoreml.l. Corollary 1.1 (Gromov[5}) There does not exist any closed exact Lagrangian submanifold in (T*(M),d(pdq)) which misses the zero section M. Theorem 1.3 Let (SU, w( = d(tA))) be the symplectization of contact manifold (U, A) and cp : L -+ (SU, d(tA)) a closed Lagrangian embedding, then [cp*(tA)] f in H* (L, R), especially HI (L) f 0.
°
For (R 2n ,wo) this is a deep theorem due to Gromov in [5]. According to [2,4]' the following notion was suggested by D. Bennequin. An (n + I)-dimensional submanifolds L of the (2n + I)-dimensional contact manifold (U,~) is called pre - Lagrangian if it satisfies the following two conditions: -L is transverse to ~;
Exact Lagrangian submanifolds
305
and can be defined by a closed I-form. For any pre-Lagrangian submanifold LeU there exists a Lagrangian submanifold i c S~U such that 7r(i) = L. The cohomology class A E Hl(L;R), such that 7r*A = [o:~li], is defined uniquely up to multiplication by a non-zero constant. Conversly, if LeU is the (embedded) image of a Lagrangian submanifold i c S~U under the projection S~U -+ U then L is pre-Lagrangian(see[2,4]). Thus with any pre-Lagrangian sub manifold LeU one can canonically associate a projective class of the form A. Theorem 1.4 There does not exist any pre-Lagrangian submanifold LeU with the canonical projective class equal to zero, especially any simply connected manifolds can not be embedded in (U,~) as a pre-Lagrangian submanifold. Let U be a smooth oriented manifold of dimension 2n - 1 and A contact form on U is a I-form such that A1\ (dA)n is a volume form on U. Associated to A the so-called Reed vectorfield X = X oX defined by
iXA == 1, ixdA == 0 and Reeb flow 1Js . Theorem 1.5 Let (U, A) be a contact manifold with contact form A and LeU a closed pre-Lagrangian submanifold, the L n 1JsL is non-empty. Proof. Since 1Js is Reeb flow we have 1J; A = A. It is well-known that symplectization of contact flow is homogenious symplectic flow (in fact Hamiltonnian flow), Theorem 1.1 completes the proof. Similar to [2], we can use the above theorems to define the contact invariants, we will discuss it in an another paper. Sketch of proofs: We will work in the framework proposed by Gromov in [5) . In Section 2, we study the linear Cauchy-Riemann operator and sketch some basic properties. In section 3, we study the space V(V, W) of disks in manifold W with boundary in Lagrangian submanifold W. In Section 3, we construct a Fredholm section of tangent bundle of V(V, W). In Section 4, we use the Gromov's trick in [5) to reduce the intersection of Lagrangian isotopy to the immersion of a fixed Lagrangian submanifold. In the final section, we use Gromov 's nonlinear Fredholm trick to complete our proof as in [5) . Note 1.1 One can easily extend the trick in this paper along with Floer's theory to build the Morse and Lusternik-Shnirelman theory since d(tA)I7r2(SU) = 0, we will discuss it in another paper, some related but more difficult results see
[4].
306
Renyi Ma
§2. Linear Fredholm Theory For 3 < k < 00 consider the Hilbert space Vk consisting of all maps u E Hk .2(D, en), such that u(z) E Rn c en for almost all z E aD. Lk-1 denotes the usual Hilbert L k _ 1-space Hk_1(D,e n ). We define an operator [} : Vp t-+ Lp by (2.1) where the coordinates on Dare (s, t) = s + it, D = {zllzl :::; I}. The following result is well known. Proposition 2.1 [5] [} : Vp t-+ Lp is a surjective real linear Fredholm operator of index n . The kernel consists of the constant real valued maps.
Let (en, (1 = -I m(·, .)) be the standard symplectic space. We consider a real n-dimensional plane Rn c R2n. It is called Lagrangian if the skew-scalar product of any two vectors of Rn equals zero. For example, the plane p = 0 and q = 0 are Lagrangian subspaces. The manifold of all (nonoriented) Lagrangian subspaces of R 2 n is called the Lagrangian-Grassmanian A(n). One can prove that the fundamental group of A(n) is free cyclic, i.e. 7r1(A(n)) = Z. Next assume (r(Z))zE8D is a smooth map associating to a point z E aD a Lagrangian subspace r(z) of en, i.e. (r(Z))zE8D defines a smooth curve a in the Lagrangian-Grassmanian manifold A(n). Since 7rdA(n)) = Z, we have [aJ = ke, we call integer k the Maslov index of curve a, we denote it by m(r), see([l D. Now let z : 8 1 t-+ Rn c en be a smooth curve. Then it defines a constant loop a in Lagrangian-Grassmanian manifold A(n). This loop defines the Maslov index m(a) of the map z which is easily seen to be zero. Now Let (V,w) be a symplectic manifold and W C V a closed Lagrangian submanifold. Let u : D2 -+ V be a smooth map homotopic to constant map with boundary aD E W . Then u*TV is a symplectic vector bundle and (UI8D)*TW be a Lagrangian subbundle in u*TV. Since u is contractible, we can take a trivialization of u*TV as q>(u*TV)
=D
x en
and q>(UI8D)*TW) C 8 1
X
en
Lemma 2.1 Let u: (D2,aD2) -+ (V,W) be a ek-map (k Then, m(uI8D) = 0
> 1)
as above.
Proof. Since u is contractible in V relative to W, we have a homotopy q>s of trivializations such that q>s(u·TV) = D x en
Exact Lagrangian submanifolds
307
and Moreover
c)O(UI8D)*TW
= S1
X
Rn
So, the homotopy induces a homotopy in Lagrangian-Grassmanian manifold. Note that m(h(O, .)) = O. By the homotopy invariance of Maslov index, we know that m(uI8D) = O. Consider the partial differential equation
au + A(z)u = 0 on D u(z) E r(z)Rn for z E aD
(2.3)
r(z) E GL(2n, r) n Sp(2n)
(2.4)
m(r) = 0
(2.5)
(2.2)
For 3 < k < 00 consider the Banach space Vk consisting of all maps u E Hk,2(D, en) such that u(z) E r(z) for almost all z E aD. Let L k- 1 the usual L k - 1 -space H k - 1 (D, en) and
Lk_dS 1) = {u E Hk-1(S1)lu(z) E r(z)Rn for z E aD} We define an operator P: Vk ~ L k- 1 X L k- 1(S1) by
P(u) = (au
+ Au, UI8D)
(2 .6)
where D as in (2.1) . Proposition 2.2 [5] a : Vp ~ Lp is a real linear Fredholm operator of index n.
§3. Nonlinear Fredholm Theory 3.1. Adapted metrics in symplectic manifold (M,w). A Riemannian metric 9 on M is called adapted (to the symplectic form w) if 9 + Aw is a Hermitian metric with respect to some almost complex structure J : T(M) ~ T(M) preserving 9 and w. This is equivalent to the existence of a g-orthonormal coframe Xi, Yi,i = 1, ... , n = dimM/2, at each point in M such that w equals E~ Xi A Yi at this point. Yet another equivalent definition reads
for all smooth functions H on M, where, recall, gradwH is the (Hamiltonian)
308
Renyi Ma
Let us show that a complete adapted metric always exists. Lemma 3.1 (Eliashberg-Gromov[3)} . Every symplectic manilold M admits a complete adapted metric g .
= (M,w)
Proof(due to [3]). The required metric will be constructed starting with arbitrary adapted metric go and applying a certain symplectic automorphism A of T*(M) to it. This A is constructed with an exhaustion of M by compact domains with smooth boundaries Si expands go transversally to all Si· Namely, we take small €i-neighbourhoods Ni C M of Si, normally (with respect to go) decomposed as Ni = Si x [-€i,€i). We denote by ~i C T(Ni ) and Vi E T(Ni ) the subbundles tangent and normal to the slices Si x t, t E [-€i, €i], respectively and take some symplectic automorphisms Ai : T(Ni ) -+ T(Ni ) preserving the decomposition T(Ni ) = ~iffivi and acting on Vi by Ai(V) = 2v. Then A is taken equal to Id outside all Ni and AIT(Ni ) =deJ AlP, where
Note 3.1 Let I(M,w)
= inf{1
r
ls2
/*wl
> 011 : S2 -+ M}
and ~ C M x B 2 (.,jI(M,w) smooth hypersurlace 01 contact type, then the argument in [8-9} along the above Lemma implies that there exist at least one closed orbits on ~. This removes the assumption in [8-9} which assume M is closed or with contact type boundary. 3.2. Formulation of Hilbert manifolds. Now let (U, A) be a contact manifold with contact form A. Let SU = (Ux)O, 00[, d(tA) be its symplectization. Associated to A there are two important structures. First of all the so-called Reed vectorfield X = X>. defined by
iXA == 1, ixdA == 0 and secondly the contact structure { = {>.
f-t ~
given by
= ker(A) C T~ and put { = ker(A). Then dA is a symplectic structure {>.
Let SU = (Ux)O,oo[) for the vector bundle { -+ ~. We choose a complex structure J for { such that gJ : dA 0 (/2 x J) is a metric for { -+ ~. As in [6], we define an almost complex structure J on VV by
J(t, u)(h, k) = (-A(u)(k), J(u)7rk 7r : T~ f-t { is the bundle projection along RX associated to A.
f-t {
+ hX(u)) and X the Reeb vectorfield
Exact Lagrangian submanifolds
309
Proposition 3.1 There exists an adapted complete metric on the symplectization S U = (UxIO, oo[, d(tA)) of contact manifolds (U, A).
In the following we denote by (V, w ) = (SU x R2,d(tA) CB dx A dp)) with the adapted metric g @go and W c V a Lagrangian submanifold which will be defined later. Let , E W a.e for x E d D and u(1) = p) v k ( v , W,p) = {u E H ~ ( DV)Iu(x)
Lemma 3.2 Let W be a closed Lagrangian submanifold in V. Then, Dk(v,W,p) = (u E H ~ ( D , V ) ) U ( XE) W a.e for a: E a D and u(l) = P) is a Hilbert manifold with the tangent bundle
here
{H"'
A~-'(u*Tv, u(gDTW,p)= - sections of (u*(TV),( u ( ~ ~ ) * Twhich L ) vanishes at 1)
(3.2) (3.3) (3.4)
Proof: See[5]. Now we construct a nonlinear F'redholm operator from vk(V,W,p) to TVk(v, W, p) follows in [5]. Let 8 : Vk(v, W, p) + TVk(V, W, p) be the CauchyRiemmann Section induced by the Cauchy-Riemann operator, locally,
for u E Vk(V, W,p). Since the space vk(v, W, p) is Hilbert manifold, T v k(v,W, P) the tangent space is trivial, i.e. there exists a bundle isomorphism
t Then the Cauchy-Riemann section 8 on TDk(V, W,p) where E is a ~ i i b e rSpace. induces a nonlinear map
In the following, we still denote a? o 8 by
8 for convenience. Now we define
Renyi M a
310
Theorem 3.1 The nonlinear operator F defined i n (3.6-3.7) is a nonlinear Fredholm operator of Index zero.
Proof. According to the definition of the nonlinear Fkedholm operator, we need to prove that u E Vk(V,W,p), the linearization DF(u) of F at u is a linear Fkedholm operator. Note that
where
with v l a ~E ( u l a ~ ) * T W here A(u) is 2n x 2n matrix induced by the torsion of almost complex structure, see [5] for the computation. Observe that the linearization DF(u) . . of F at u is equivalent to the following Lagrangian boundary value problem
One can check that (3.10-11) defines a linear Fkedholm operator. In fact, by proposition 2.2 and Lemma 2.1, since the operator A(u) is a compact, we know that the operator F is a nonlinear Fkedholm operator of the index zero. Definition 3.1 A nonlinear Redholm F : X + Y operator is proper if any y E Y , Fdl(y) is finite or for any compact set K C Y , F-'(K) as compact in X. Definition 3.2 deg(F, y) = #{F-'(y)}mod2 is called the fiedholm degree of a nonlinear proper Fredholm operator(see[5,11]). Theorem 3.2 Assume that the nonlinear Redholm operator F : Vk(v, w,p) -, E constnlcted i n (3.6-7) is proper. Then,
V be a J-holomorphic disk with boundary Proof: We assume that u : D u(~Dc ) W . Since almost complex structure ?tamed by the symplectic form w , by stokes formula, we conclude u : D 2 -+ w is a constant map. Because u(1) = p, We know that F-'(O)
Exact Lagrangian submanifolds
311
DF(p) of Fat p is an isomorphism from TP'D(V, W,p) to E . This is equivalent to solve the equations
=f
(3.12)
VlaD C TpW
(3.13)
8v 8s
J8v
+ 8t
By Lemma 3.1, we know that DF(p) is an isomorphism. Therefore deg(F, 0) = 1.
Corollary 3.1 deg(F, w)
= 1 for any wEE .
Proof. Using the connecticity of E and the homotopy invariance of deg.
§4. The non-properness of the Fredholm Operator We shall prove in this section that the operator F : 'D -+ E constructed in the above section is non proper. Following Gromov [5], by using the Lagrangian isotopy F' to construct a Lagrangian immersion of L x S1 into (V x C, w E9 wo), and relate the double points of the later with the points of L n II (L) more precisely we have
Lemma 4.1 Let (SU,w) = d(t>.)) be the symplectization of contact manifold (U, >') , F' be an exact isotopy and let the submanifolds W' = F'(W' x t) c (SU,d(t>.)), t E [0,1] be exact. Then there exists an exact Lagrangian immersion F* : W' x S1 -+ SU X C such that the intersection Wo n W{ the set of double points of F*(W' x S1). Proof. See [5,2 . 3B~]. Now let c E C be a non-zero vector. We consider the equations
u=
(U1' U2) :
D -+ SU x C
(4.1)
= 0, 8U2 = c
(4.2)
UlaD: 8D -+ W
(4.3)
8U1
(4.4) Since W is a closed manifold in SU x C we know that
(4.5) here C1 depends only on W . Since every solution harmonic, it satisfies
U2
to the equation
8U2
= c is
Renyi Ma
312 and so there is no solution U = (Ul, U2) : (D2, 8D 2) --+ (SU equations for large Ilel!. i.e. we have proved
X
C, W) to this
Theorem 4.1 The Fredholm operator F : Vk(V, W,p) --+ E is not proper.
Proof. By Theorem 3.1 and 3.2, we know that the index of F is zero and deg(F) = 1, then the above argument show that F is not proper.
§5. The existences of holomorphic planes and periodic solutions In this section, we use the Sacks-Uhlenbeck-Gromov's trick to prove the existence of J-holomorphic disk with boundary in W if WE SU x C is Lagrangian submanifold. Now let e E C be a fixed non-zero vector satisfying Theorem 4.1. We consider the equations
(5.1)
U = (Ul' U2) : D --+ SU x C 8Ul
= O,8U2 = se
(5.2) (5.3) (5.4)
uI8D : 8D --+ W
In order to get the estimate of energy on solution u, we consider (V, W, d>.) = (SU x C, dN EB dx /\ dy) where SU is the symplectization of contact manifold (U,N). Now we recall the symplectization of (U,>.) is (SU,d(t>')) = ((Ux)O,oo[),d(t>.)) and put ~ = ker(>.). Then d>' is a symplectic structure for the vectorbundle ~ --+ U . We choose a complex structure J for ~ such that gJ := d>.o (Id x J) is a metric for ~ --+ U. As before we define an almost complex structure J on SU by
J(t, u)(h, k)
= (->,(u)(k), J(u)1fk
+ hX(u)),
(5.5)
where 1f : TU --+ ~ is the bundle projection along RX --+ U and X the Reeb vector field associated to N . We define a complete metric 9 on SU by
< (hI, kd, (h2' k2) >= hlh2 + >'(kd>'(k2) + gJ(1fk 1 , 1fk2)'
(5 .6)
By Lemma 3.1, we have a complete adapted Riemann metric on SU. So, we have an almost complex structure J = J 1 EB i on SU x C and adapted complete metric 9 = gl EB go on SU x C . Now for U = (Ul, U2) : (D2, 8D2) --+ (SU X C, W), define
r
8u8u 8u8u E(u) = JD(g(8x , J 8x)+g(8x , J 8x))dcr
(5 .7)
Exact Lagrangian submanifolds
Lemma 5.1 Let (V, W)
u
313
= (SU x C, W), J,g
as above and
= (Ul, U2) : D -t SU X C aUl = 0, aU2 = sc UI8D:
aD -t W
(5.8) (5.9) (5.10) (5.11)
Then, we have the following estimates
(5.12) here
C3
depends only on c, symplectic form w, W .
Proof. See (5) . Lemma 5.2 Let (W, V) = (SU x C, W), J, g as above and U = (Ul,U2) : D -t SU aUl
X
C
(5 .13)
= 0, aU2 = sc
(5.14)
aD -t W
(5.15)
UI8D:
(5.16) Then the image of U SUxC.
= (Ul, U2)
: (D2, aD2) -t (SU x C, W) is bounded in
Proof. That u2(D2) is bounded in C follows as above since W is closed and U2 is harmonic. Since Ul is J -holomorphic and Ul (aD) is bounded and its energy or area is bounded by the above Lemma, then the monotonicity of minimal surfaces concludes that Ul can not go to infinity. Theorem 5.1 There exists a non-constant solution U : (D2,aD2) -t (SU x C, W) of the partial differential equations Us
+ J(u)Ut = 0
(5.17)
where
(5.18)
Proof. By the above Lemma, we know that all arguments in [5] for the case V' is closed in (V' x C, W) can be extended to our case V' = SU is not closed since the images of maps remains in a bounded set.
314
Renyi Ma
Proof of Theorem 1.1. By the assumption of Theorem 1.1, we know that the Lagrangian submanifold W in SU x C is embedded and exact if Theorem1.1 does not hold. Then for large vector c E C we know that the nonlinear Fredholm operator or Cauchy-Riemann operator has no solution, this implies that the operator is non-proper. The non-properness of the operator implies the existence of J-holomorphic disk with boundary in W which contradicts the fact that W is exact since J -holomorphic disk has positive energy. For more detail, see [5].
References [1] Arnold, V.& Givental, A., Symplectic Geometry, in: Dynamical Systems IV, edited by V. I. Arnold and S. P. Novikov, Springer-Verlag, 1985. [2] Eliashberg, Y., New invariants of open symplectic and contact manifolds, Journal of AMS. 4(1991), 513-520. [3] Eliashberg,Y.& Gromov, M., Lagrangian Intersection Theory: FiniteDimensional Approach, Amer. Math. Soc. Transl. 186(1998) : 27-118. [4] Eliashberg,Y., Hofer,H., & Salamon,S., Lagrangian Intersections in contact gemetry, Geom. and FUnct. Anal., 5(1995): 244-269. [5] Gromov, M., Pseudoholomorphic Curves in Symplectic manifolds. Inv. Math. 82(1985), 307-347. [6] Hofer, H., Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjection in dimension three. Inventions Math., 114(1993), 515-563. [7] Klingenberg, K., Lectures on closed Geodesics, Grundlehren der Math. Wissenschaften, vol 230, Spinger-Verlag, 1978. [8] Ma, R., A remark on the Weinstein conjecture in M X R2n. Nonlinear Analysis and Microlocal Analysis, edited by K. C. Chang, Y. M. Huang and T . T. Li, World Scientific Publishing, 176-184. [9] Ma, R., Symplectic Capacity and Weinstein Conjecture in Certain Cotangent bundles and Stein manifolds. NoDEA.2(1995):341-356. [10] Sacks, J.& Uhlenbeck, K., The existence ofminimaI2-spheres. Ann. Math., 113(1980), 1-24. [11] Smale, S., An infinite dimensional version of Sard's theorem, Amer. J. Math. 87(1965): 861-866.
Low dimension anomalies and solvability in higher dimensions for some perturbed Pohozaev equation G. Mancini* Dipartimento di Matematica, Universita di Roma Tre Abstract In this talk, we will present a new solvability condition in higher dimension for a Brezis Nirenberg type equation. This result, which we believe to be sharp in some sense, has been obtained in collaboration with Adimurthi.
1
Introd uction
Very surprising "low dimension phenomena" have been observed by Brezis and Nirenberg in their pioneering work [BN] concerning perturbations of the "Pohozaev equation" -~u
=U
u>O u= 0
~ N - 2
+ p(u)
in 0
inO in 80
Here, 0 C !RN , N ~ 3 , is a smooth bounded domain and p is sub critical , with p' (0) = O. N Dealing with pes) = s1i1=2. Brezis and Nirenberg discovered an anomalous behaviour in dimension 3: (i)if N > 3, (P) has a solution '
4-N
(*) lim f -2-
1
.-to
where P(s)
=
J; p(t)dt.
Ixl<..L -../<
1
P(( f
+
X
Of course, pes)
N = 3. This example shows two facts :
anomalies may occurr in dimension 3 - condition (*) is sharp in dimension 3. 0*
N-2
1 12 )-2-) = +00
Supported by M.U .R.S.T.
315
=
SN":2
does not satisfy (*) if
G. Ma.ncini
316
In view of this example, two related questions naturally arise: - is condition (*) sharp also in higher dimensions? - do anomalies occur also in higher dimensions ? We also refer to [PS], [BG] for related problems and interpretation of " low dimension phenomena" In a recent paper [AY], Adimurthi and Yadava gave an adfirmative answer to the question above in dimensions 4, 5, 6: they exibited a class of compactely supported perturbations PN such that (k) if N = 3, 4 and P E PN, there are anomalies (i.e. there are solutions on Br iff r is large) (kk) if N = 5,6, and P E PN and , in addition, J P( IxI L2 ) = 0, then there are anomalies. (kkk) if N:2: 7, and P E PN, then (P) is solvable on any ball. Of course, in cases (k) - (kk) condition (*) is violated: in case (k), because obviously J P( ~) < 00, while, in case (kk) , because
J P( ~) = 0 implies J1xl::;-}. P(C+ixI 2 (2"2) = O(f)
(see Lemma 4.1 below) .
In the attempt to extend the existence result in (kkk) to more general perturbations and domains, Adimurthi and myself [AM] realized that condition (*) is not sharp in dimension N :2: 7. To be more precise, let us first give a closer look to condition (*) , limiting ourselves, for sake of simplicity, to perturbations p in the class P := C8"((O, 00)). If we denote, for a given pEP
h
= h(p):=
1
RN
dk 1 [-dkP(~)](lxI2)dx
t
t-2-
it is easy to see that (j) If h = 0 Vk ,orlo = It ··· = h-l and h < 0 for some k, (*) fails in any dimension N (jj) If 10 = It = .. . = h-l = 0, and h > 0 for some k = 0,1... then (*) holds iff N > 2k + 4.1n fact, if 10 = It = ... = h-l = 0 then limHo f 4-;N ~xl::;* P(C+ixI 2 ) N;-2)= . 4±2k-N f dk 1 N-2 hm<-+of 2 JRN wP(C+1xI2)-2-) k
2
f d P (( 1 ) N- ) an d JRN w ~ -2-+<-+0 h (Let us note, incidentally, that if p belongs to the Adimurthi-Yadava class PN , then Io(p) = 0 =} ll(p) > 0, and this explains the "surprising" existence result (kkk)) . In particular:
Some perturbed Pohozaev equation
317
10 = 0 and 0 > II = _N:;2 f p(~)W On the other hand , we will prove that, if
(**) 10 = 0 and
J1\7
HI 2 -
then(P) has a solution on any solution of
~
=}
(*) fails '
J Ixl~-2) Ix~N > p(
n if N 2::
0
7. Here, H is the finite energy
-~u = p( Ixl~-2 ) in !RN So, we see that, in case 10 = 0, a much weaker condition than (*)
(1.1)
insures, in dimension N 2:: 7, the solvability of (P) on any domain, i.e. (*) is not sharp in dimension N 2:: 7. Of course, all this does not answer the question whether the anomaly described above occurs in dimension N 2:: 7. Even if we do believe that such anomalies occur in any dimension (i.e. "anomaly" is not a low dimension phenomena!) , we content ourselves with the weaker conjecture N
if (P) has solutions with energy smaller than ~"2 (S := best Sobolev constant in the embedding HI --+ L J~2 (!RN)) on any small ball B r , then
In other words , we suspect that a weak inequality in (**) is necessary for "low energy" solutions to exist on any small ball.
2
The main result and the variational approch
For sake of simplicity we will assume p E Cgo((O,oo)) . Let H be the unique solution in V 1 ,2 of
Theorem 2.1 If f P( ~)
>0
and N
k
2:: 5 , (P) has a solution.
If.f P(~) = 0 and f I\7HI2 > f p(~)W' then if N 2:: 7, (P) has a solution '
G. Mancini
318
By standard arguments, a non zero critical point of 1 E(u) := 2"
J
N - 2 j" (U+)N-2 2N -"2N
lV'ul 2
-
j"P(u+) U
E
1 ) Ho(O
is a solution of (P). The energy functional E enjoies the mountain pass geometry, and, again by standard arguments, a mountain pass level is critical
it"2 N
provided it is strictly smaller than (S := best Sobolev constant). In fact it is enough to prove that, for some V E HJ (Br) , Br C 0, it results
SIf
maxE(tV) < N
(2.1)
t~O
where E(u) = ~ I lV'ul 2 - ~-;/ I lui J~2 - I P(u) , u E HJ(Br ) and P is the even extension of p(t)dt, s 2: O. This is because, after extending V equal to zero outside B r , it results E(tIVD ~ E(tV).
I;
3
Choise of the test function and basic estimates
To get (2.1), we will choose V satisfies
-~ V,
Here UE(x) = U(x)
=
= V"
for
f
.!Y.H.
ut- + p(U,) 2
suitably small, where
on Br C 0
V.
E HJ(B r )
(3.1)
N-2
C-2-U(~)
= (1~~12(;2
Recall that -~UF =
,
= IN(N - 2) ut- in !)(N, and CN
.!Y.H. 2
In what follows, we will always assume f < ~r2, where § = c"N1J N"-2 and J < 1 is such that p(s) == 0 for s ~ J. We will denote WE := V.N-2 - U" so that -~w, = p(U,) in Br and W F ==-bE on 8B r where b, = (,&:{:;2) -2-. N-2 We will also write sometimes eN instead of cT
Some perturbed Pohozaev equation
319
Lemma 3.1 Let V. be given by (3.1). Then E(V,)
~ ~!f + +[0(1) + CN N;2 2
N
- 1 +;(1) where 0(1) are small in
( l'RN
U~]( ~ )N-2 _ r
LrlV'w,1 2- Lr
P(U, )
(3.2)
uniformly with respect to r E (1 , 00)
f
Proof. Multiplying (3.1) by V, and integrating by parts, and using the inequality 2 N..:ilL 2
Iv.l'iV-2 - ut-
~ ~~2
..:ilL
ut-
2
w" we get, by straightforward computations:
N ~ 2 E(V.) ~ if' - H!t"'I~r ut- w, - p(U,)U, + p(U,)w,]- fBr P(U,) - fBr X [O,l](1 - t)p'(U, + tw,)W;
Now, from -6.U,
{ lBr
~
= ut- 2 ,
-6.w,
= p(U,), and Green formulas , we get
U,~w, = ( p(U, )(U, - w,U,~), lBr
{
11"'I~r and hence, since f
p(U, )w,
= ( lV'w,1 2-
~2 ut~
N-2
~
~ ~' + eN (~)N-2 N
-~ { lV'w,12- (
(3.3)
f'RN U N-2 :
f -2-
N
E(V.)
w,p(U,)
lBr
2
r
U~ +b, ( p(U,)lBr
+ {
(l-t)p'(U,+tB,)w; (3.4) lBr x[O ,l] To complete the proof of the Lemma, we just use (3.4) and the estimates gathered in the following
11"'I~r
2 lBr
P(U,)
{ l'RN
Lemma 3.2
+ 0(1)] ()[f = 0 1 JI"'I~r I'"v w, 12
(i)
!t"'I
(;,;) ..
f - Iw,1 ..:ilL)N-2 (JI"'I
(iii) (iiii)
sUPr>l II W, IILOO(B r )= O(f2) I fBr:[O,l](t - l)p'(U, + tw,)w;1
-
1
and all the estimates are uniform in r
+ (')N-2] r
= O(I)(JBr lV'w,1 2+ (~)N-2) ~
1
G. Mancini
320 Proof To see (i), we just make the change of variable y = fBr
p((.2~1~12 (;2) = Elf
because ~
s;
r2 , Iyl ;:::
flRN
1
C '2 x
:
P((.;~12 (;2 )dy
:it ~ (';~12)-2- s; (~)-2- ::; (JCN) N-2
N-2
N-2
'
2 = J ~
P((.;~12(;2) = 0 Hence I p((.2~1~12 (;2)
= Elf(o(l) + flRN p( !yIH-2)) where 0(1) is uniform with respect to r ;::: 1 As for (ii), it follows by Sobolev inequality using w. == -b. on aBr :
(3.5) where WN := fBI dx. To see (iii), we can use the representation formula (recall that p(U.) == 0 outside B y'f C Br) :
w.(x)
+ b. = Clll -WN IYI~y'f p(U.(y))( Ix -
1 Y
IN-2 - 1-
1
I - )dy x- YN 2
(3.6)
where x is any point on aBr' Now, Ixl ;::: 2y'f ~ Ix - yl ;::: y'f ~ ~YI~y'f Ix - y12-N ::;
(~(;2 (~)lfWN = WN~' while Ixl s; 2y'f implies ~YI~y'flx _ y12-N r IZ < Jlyl9y'f _
1
2 N --
9 •' zWN"6
So, for any x we obtain from (3.6) Iw.(x)1 s; (~(;2 + dNE (dN just depending on Nand p) that is II w. 1100= O(d) uniformly for r ;::: l. Finally, by Holder inequality and (ii), we have
By dominated convergence, fol (fB r Ip'(U. +tw.)llf)k -+HO 0 for any given r, because II w. IIL<»(Br)-+'-+o 0 for every r by elliptic theory. Furthermore, by (iii), if r ;::: 1 , p'(U. + tw.)(x)) == 0 for Ixl ;::: y'f and E < EN, for some EN
321
Some perturbed Pohozaev equation
depending only on N and not on r, and hence we get (fB Ipl(U. +twf )I~)1r 2
! II pi 1100 wt 4
:::;
r
E.
This complets the proof of (iiii) .
Higher order estimates and the proof of the Theorem
We first estimate the last two terms in (3.2) . Lemma 4.1
(i) (ii)
(iii)
Furthermore, all estimates are uniform in r E (1,00) . Proof (i) Let ti\(x) := c1w.(J€x) in B...!:-, so that .Ji
_
-6w. =p((
eN!:!..=2
11 ) E +X 2
2
. )
III
B.L . .Ji
clearly W. - H. == const on B.L, and hence .Ji
IBr IVw.1 2 = 0(1)), where
~
E 2
IB IVw,1 2 = E ~J 2 Br IVH.I 2 = E ~(II VHO 12 + 7.
2
7-
-6Ho = p( IX~:-2 ) in RN
and 0(1) is uniform with respect to r in [1,00). Then (i) follows from ~
I IVHol2 = eN IIVHI 2 • 2
G. Mancini
322
y = eh
(ii) After the change of variable
we have
where the second integral is in fact taken over B l , because
f. ::;
./6
8r2 . First
(4.1) If this integral vanishes,
limHo
~ ~YI~l6 P(C':~12(2"2) ./6
=
N2
N
= limHO ~YI~~ CN -,t p(C':~12)-T-)C+iYI2)2 = = _N;2CN I91N p(lyfH- 2 )W = _N;2CN I91N p(rzp6-)W 2
where the limit is obviously uniform in r Finally, we prove (iii) . We have limHo elf
~
1.
IBr U
, uniformly in r ~ 1, oo and I91N ~p(IYIH-2) = NWN Io PP(pW!-2)dp =
= I91N lyfH-2P( lylH-2 )dy --1
= :'':.2 I91N P( IxIH-2) = 0 by integration by parts. So, we obtain, as above
IBrU
-
N;2 CN
I W[P(lxIH- 2)+
Proof of the theorem We have to show how to derive (2.1) from Lemma 3.1 and 4.1. We will limit ourselves to the "vanishing case" I P( ~) = o. Inserting the estimates (i) - (ii) of Lemma 4.1 in (3.4), we get
~ 2
N -c -f.~ 2 [0(1) 2
+
/
IVHI 2 - -1 N
/ -1p ( - _ 1 )] N N 2
Ixl
IxI
-
(4.2)
323
Some perturbed Pohozaev equation
where 0(1) are uniform in r ~ 1 and aN does not depend on r. Since, by assumption, II := J IV'HI2 J Wp(~) > 0, we see from (4.2) that for € smaller than some €(r)
11
S E(V:) < < - N
N
2
-
€
N
-
2
h
6-N eN (_€-2 - - ) 2 rN - 2
and in fact €(r) can be taken independent on r if r Now, (2.1), and hence the result, follows from the
ClaiIn Let E(t<~) = maxt>O E(tV<) ,
€
(4.3)
~
1.
< €(r). Then
where 0(1) is uniform with respect to r E [1,00).
Assuming the claim, we have
uniformly in r ~ 1, and hence, from (4.3): N
S2 maxE(tV:) < t>O < N for
€ ::;
€(r) \:Ir, or
€ ::; €o :=
€
N
-
2
h
6-N
(_€-2- -
4
infr~1 €(r)
> 0, if r
if N ~ 7 then, \:Ir > 0 : mFE(t~) <
+
1 eN --) rN - 2
~ 1. So
sIf
2
SIf
if 3 < N < 6 then maxE(tV<) < for t>O N
€ ::; €o
for
€
< €(r)
and r ~ r(€)
G. Mancini
324
Proof of the Claim Let us first observe that
(4.4)
!
2N
1V,1i¥=>
(f)N 2)1 = S2N + O(I)(f ~ 2 + ~ - "2
In fact, I IV'V, 12 = I IV'U,1 2 + I lV'w, 12 + 2 I V'V, V'w, O(I)U' )N-2 + O(f¥) + 2 IB p(U,)V, r
andIIp(U,)Y.I~LNf
~ 4
(4.5)
=
y't I 2N ) 2N, N-2 £or some pure constant L N· ( 1Y.1i¥=>
2N° 2N 4 Also, I IV, IN-2 = I ut -2 + ;::'2 I IU, + tw, Ii¥=> (U, + tw,)w, 2N N±2 I (1- t)IU +tw IN~2W2 = sIf +O(I)(f¥ + (~)N-2)~ + N-2 N-2 B~x[O,11 " , .r .
by Holder inequality, Lemma 3.2-ii and Lemma 4.1-i. Also, all estimates m (4.4) (4.5) are uniform in r 2 1. Now, from (4.4)-(4.5) and
we get
N)~
because I Jp(t,Y.)Y.1 ~ LNIY.I,.J~2(f2 2N
This proves the claim, because It, - 11 ~ N;-2I t
4
t- 2 -
11.
References [AM] Adimurthi and Mancini, A sharp condition in higher dimensions for a Brezis Nirenberg type equation, in preparation [AY] Adimurthi and Yadava, Critical Sobolev exponent problem in a ball with nonlinear perturbation changing sign, Advances in Differential equation 2,(1997), 161-182.
Some perturbed Pohozaev equation
325
[BG) Bemis F. and H. -Ch. Grunau, Critical exponents and multiple critical dimensions for polyharmonic operators, J.Differ Equations 117, (1995), 469-486. [BN) Brezis H. and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical exponents, Comm.Pure. Appl. math.36 (1983), 437-477. [PS) Pucci P. and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures at Appl. 69 (1990), 55-83.
A NEW PROOF OF A THEOREM OF STROBEL Paul H. Rabinowitz· Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA E-mail: rabinowi
1. Introduction In his doctoral dissertation [1], among other things, Kevin Strobel proved some basic results on the existence of heteroclinic orbits for a family of Hamiltonian systems. In particular) he considered the Hamiltonian system (HS)
q + Vq(t, q) = 0
where V satisfies (V d
V E C2(~
(V 2 )
V(t,O)=O>V(t,x),
X
~n,~) and is I-periodic in t and in q1, ... , qn, xE~n\zn.
Equations of this form arise in particular in making simple models of the motion of a multiple pendulum. By (V 2 ), all x E zn are equilibrium solutions of (HS) and it is natural to ask whether there are heteroclinic solutions of (HS) joining a pair of such equilibria. As a first result in this direction, Strobel showed: Theorem 1.1 [1] Suppose V satisfies (V1)-(V2)' Then for any distinct pair ~,
'T} E
zn,
there is a heteroclinic chain of solutions of (HS) from
~
to'T}.
More precisely Theorem 1.1 says there is an e EN and solutions Q1"'" Ql of (HS) with Q1(-00) = ~, Qi(oo) = Qi+1(-00) E zn, 1 ::; i ::; e - 1, and Ql(oo) = 'T}, i.e. Qi is heteroclinic from Qi( -00) to Qi(oo). Theorem 1.1 "This research was sponsored by the National Science Foundation under grant # MCS8110556 .
326
A new proof of a theorem of Strobel
327
was proved using a minimization argument. Set L(q) Lagrangian associated with (HS) and let I(q) =
Hil2 -
V(t, q), the
L
L(q)dt,
the corresponding functional. Define
Thus r(~, 1]) is a class of candidates for heteroclinic solutions of (HS) joining ~ to 1] . Let c(~, 'T}) =
inf
qH(€,lI)
I(q) .
(1.1)
Analyzing a minimizing sequence for (1.1) shows a subsequence "converges" to a heteroclinic chain Ql, .. . ,Ql where Ql(-OO) =~, Qi(OO) == ~i' 1:::; i:::; l, ~o =~, ~l
= 'T}, l
c(~, 'T}) =
L C(~i-l' ~i)
(1.2)
i=1
and for 1 :::; i :::; l, (1.3) Earlier work in this direction was done by Bolotin [2] - see also Kozlov [3] and [4]. With Theorem 1.1 in hand, one can ask when there is an actual heteroclinic rather than a heteroclinic chain joining ~ and 1] . One cannot expect this to be the case without further restrictions on ~ and 'T} or on the basic chain joining them. E .g. suppose n = 1 and V = V(q) = 1 - cos 211"q, the simple pendulum potential. Then a phase plane analysis shows heteroclinics from ~ to 'T} exist if and only if 'T} = ~ ± 1. In the setting of Theorem 1.1, assume that Ql, .. . , Ql are isolated solutions of (HS) in the following sense: There is a v > 0 such that whenever v E r(~i-l' ~i)' 1:::; i:::; l, with I(v) = C(~i-l'~i ) and w is a solution of (HS) with Ilw - vllw1 .2(R,lRn) :::; v, then w = v. Then Strobel showed:
(1.4)
328
Paul H. Rabinowitz
Theorem 1.2 [1] If V satisfies (V1)-(V2) and (1.4) holds, then there are infinitely many actual solutions of (HS) heteroclinic from ~ to'TJ. The solutions are characterized by the amount of time they spend near the points ~i, 1 :=:; i :=:; £ - 1.
The arguments that Strobel employed to prove Theorem 1.2 were in the spirit of delicate variational deformation methods developed by Sere [5] and others, e.g. [6]-[7] to obtain so-called multibump solutions of Hamiltonian systems. More recently, Bertotti and Montecchiari [8] obtained an analogue of Theorem 1.1 when V depends on t in an almost periodic fashion and very recently Alessio, Bertotti, and Montecchiari [9] found an analogue of Theorem 1.2 for a class of slowly oscillating (in time) potentials. That the potential is slowly oscillating allows one to avoid the non degeneracy condition (1.4). See also [10] in this regard where a related result was obtained by different arguments. The goal of this note is twofold. First Strobel's result will be improved by weakening (1.4). Indeed the study of the nondegeneracy condition here is of independent interest. Secondly a new and very elementary constrained minimization argument will be used to prove the generalization. The new existence proof was motivated in part by recent work of Calanchi and Serra [11] and by [12] and [13]. The non degeneracy condition will be studied in §2 and in §3 the new existence proof will be carried out for the special case of a two chain where the main features are already present. The more general result and some extensions will appear in [13].
2. The All or Nothing Lemma In this section, the degree of degeneracy for minimizers of (1.1) will be studied in the simplest setting. Thus suppose that a, b E zn and Theorem 1.1 yields a solution, Q, of (HS) that is heteroclinic from a to b with I(Q) = c(a, b). Suppose further there are no heteroclinic £ chains Q1, ... , Q l from a to b with
e
L
I(Qi)
= c(a, b)
1
with £ > 1, i.e. we only have the simplest possible type of heteroclinic connection between a and b. For any k E Z, set TkQ(t) == Q(t-k) . Then TkQ E r(a. b) and by (Vd, I(TkQ) = I(Q). Thus all such time translates of Q are also minimizers of I in r( a, b) . Of interest here is the nature of the set of such minimizers of!.
A new proof of a theorem of Strobel
329
Let
S(a, b) = {q(O) E IRn I q E r(a, b) and I(q) = c(a, b)}. By the above remarks, q(O) E S(a, b) implies q(k) E S(a, b) for all k E Z . Therefore a, bE $(a, b). Let Ca(a, b) and Cb(a, b) denote respectively the components of $ containing a and b. Then somewhat surprisingly we have Lemma 2.3 (All or Nothing) : Either
(i) Ca
= Cb,
or
(ii) Ca = {a} and Cb = {b}. Remark 2.4 Observe that (i) occurs if V = V(q) since then I(TlJq) = I(q) for all () E IR and q E r(a, b). Alternative (ii) is surely true generically but it is difficult to give a concrete example of where it holds since the sets Ci , i = a, b, cannot be written down explicitly. However we note in passing that if the period, 1, of V in t is replaced by T, then for large T and V(t, q) =
Proof of Lemma 2.3: Set
K(a, b) = ((q(O),q'(O)) I q E r(a, b) and I(q) = c(a,b)}. The proof consists of four main steps: (A) K and S are bounded sets (in IR2n and IRn respectively), (B) K = K U {(a,O)} U {(b,O)}; $ = S U {a} U {b},
(C) The projection map P : K -+ $, (q(O),q'(O)) -+ q(O) is a homeomorphism (D) A dynamical systems argument to complete the proof. Proof of (A): It suffices to show that there is an M >
°
such that (2.1)
for all q E r(a, b) such that I(q) = c(a, b) . Indeed if (2.1) holds, (HS) provides LOO bounds for ij and interpolation inequalities then imply uniform bounds in C 2 (IR, IRn) for such q. To verify (2.1), let (2.2)
Paul H. Rabinowitz
330
where Br(z ) = {x E IRn Ilx - zl < r} . If q E r(a,b) and q(t) E IRn\Bk(:~n) for t E [<1, s], it is not difficult to show - see e.g. Lemma 3.6 of [14] - that
L S
L(q)dt ;:::
V2{3G) Iq(s) -
(2 .3)
q(<1)I ·
Thus q E r(a, b) and I(q) = c(a, b) imply that
3(c(a, b)
+ lal) + 2;: :
V2{3( ~) IlqIILOO (IR,lR
n )
(2.4)
and (2.1) is satisfied.
Proof of (B) : Let z E K\{(a, 0), (b,O)} . Then there is a sequence in K : (qm(O),q~(O)) --+ z. The form of I implies (qm) is bounded in Wl~;(IR,IRn) and by (A), (qm) is bounded in C 2 (IR, IRn) . Hence, along a subsequence, qm converges weakly in Wl~'; and in Cl~c to a solution, q* , of (HS) with initial data z. Moreover I(q*) :::; c(a,b) . Since I(q*) < 00, Proposition 3.11 of [14] shows q*(±oo) E zn. If q*( -00) 1= a or q*(oo) 1= b, standard arguments show q* is a piece of a heteroclinic i-chain {PI, . .. ,pe} of solutions of (HS) connecting a and b. But by the choice of a and b, there is no such i -chain joining a and b except for i = 1. Thus q*( -00) = a and q*(oo) = b. Hence q* E r(a, b) and I(q*) = c(a, b) . Consequently z E K(a, b) and (B) has been verified. Proof of (C) : Given (B) , once it has been shown that P is I-Ion K , that Pis a homeomorphism easily follows . To prove that P is 1-1, a "curve shortening" argument will be employed. If P is not 1-1, there exist P 1= q in r(a, b) with
I(p) = I(q) = c(a, b)
(2.5)
and p(O) = q(O). Then
(2.6) for otherwise, say
[~ L(p)dt < [0
L(q)dt.
(2.7)
L(q)dt.
(2.8)
00
Then by (2.5) ,
1
00
1
00
L(p)dt >
Define
r(t) = p(t) {
(2.9) = q(t)
t;:::O
A new proof of a theorem of Strobel
331
Then r E r(a, b) and by (2.7)-(2.8),
I(r)
= [°00 L(p)dt +
100
L(q)dt < c(a, b),
(2.10)
a contradiction. With (2.6) and therefore
1
00
L(p)dt
=
100
(2.11)
L(q)dt
in hand, define r by (2.9). Then by (2.6) and (2.11), I(r) = c(a, b). Moreover r E r(a, b). Consequently r is a solution of (HS). But then the basic uniqueness theorem for ordinary differential equations implies r(t) == p(t) == q(t). Thus P is 1-1.
Proof of (D): Suppose Lemma 2.3 is false. Then Ca f. Cb and either Ca f. {a} or Cb f. {b}. The argument is the same in either case so suppose Cb f. {b}. Let k E Z and define fk on S by fk(q(O)) = q( -k). By (C), q(O) determines q E r(a, b) uniquely so fk is well defined and continuous from S to S. Therefore fk(Cb) is compact and connected and fdb) = b. Hence (2.12) since Cb is the largest such subset of S with these properties. Note that since (i) does not hold, the distance from a to Cb , (2.13) But by the definition of
/k, as
k -+
00,
fk(q(O)) -+ a. Hence (2.14)
as k -+
00,
contrary to (2.12)-(2.13) .
3. The Existence Theorem In this section, the main existence result will be stated and a proof given for the simplest case of a 2-chain. The general case and some extensions to other situations such as infinite chains can be found in [13]. To formulate matter precisely, suppose ~ f. T) E and there is a heteroclinic i-chain of solutions of (HS), {Ql, ... ,Qd, joining ~ and T) . Set
zn
Paul H. Rabinowitz
332
Qi(OO) = ~i' 1 :S i :S e, ~o = ~, and ~i = 'T} . Further assume {QI,"" Qil is a minimal heteroclinic chain in the sense of §2, i.e. i
c(~,'T}) =
i
2: J (Qi) = LC(~i-I'~i) i=l
(3.1)
i=l
and there is no heteroclinic j-chain of solutions {UI,' .. ,Uj} joining ~i-l and ~i with j > 1 and 2:{ J(u p ) = C(~i-l'~i)' Let
Assume for 1 :S i :S
e
(*) S(~i-l'~i) nC~i_l(~i-I'~i) Hence by Lemma 2.3,
=¢
The main theorem can now be stated: Theorem 3.5 Assume (VI)-(V2) and (*) hold. Then (HS) has infinitely many solutions heteroclinic from ~ to 'T} and distinguished by the amount of time they spend near ~i' 1 :S i :S e- 1. Remark 3.6 Note that if (*) fails to hold, there is a continuum of heteroclinic solutions of (HS) joining ~i-l to ~i for some i. Thus (HS) always has infinitely many distinct heteroclinic solutions. Of course in the autonomous case, this family of solutions may just be the set of phase shifts of a single solution:
{reQ
I () E ~}.
Theorem 3.5 is a consequence of a more precise result. To formulate it, for < p« r « 1 = inf n Ix - YI· By (*) and Lemma 2.3, there are neighborhoods Oi I of ~i-l
X,Y E ~+, x « y means x is small compared to y. Let 0 x#vEZ
and
Oi ,2
of ~i' 1 :S i :S
'
esuch that j = 1,2.
Let m E ZU . Define
A new proof of a theorem of Strobel
333
Set
bm = inf I(q) .
(3.2)
qEX m
Note that it is only the difference in m i +l - mi, 1 :::; i :::; f -1, that is important in the definition of bm , i.e. if i* = (i, ... , i) E Z2l, (3.3) Now we have Theorem 3.7 For mj+l - mj sufficiently large, 1 :::; j :::; 2f - 1, there is a Qm E Xm such that I(Qm) = bm . Moreover Qm is a solution of (HS) heteroclinic from ~ to TJ with
j
IQm(t) - ~I :::; r,
t E (-oo,md,
IQm(t) - TJI :::; r,
t E
IQm(t) - ~il
[mu,oo],
(3.4)
:::; r,
The proof of Theorem 3.7 when f = 2 (and m E Z4) will be carried out. The general case involves the same ideas but is technically more complicated. For notational simplicity set 6 = (, 0 1 ,1 = O~, 0 1 ,2 = 0( , O2 ,1 = D(, and O 2 ,2 = D1/' The proof consists of several steps: (A) There exists a Qm E Xm such that I(Qm) (B) For any
€
> 0 and m2 -
= bm and
(3.4) is satisfied,
ml, m4 - m3 sufficiently large, bm :::; c(~, TJ)
+ €,
(C) An auxiliary problem which determines the choice of € , (D) The choice of m3
- m2,
(E) The completion of the proof. Proof of (A) : Let (qk) be a minimizing sequence for (3.2). Then the form of I implies (qk) is bounded in ~~~(]R, ]Rn) . Therefore as in §2, along a subsequence, qk converges weakly in Wl~~ and strongly in Lk;'c to Qm E E with Qm(±oo) E Qm otherwise satisfying the constraints of X m , and I(Qm) :::; bm . The first inequality in (3.4) will be verified next. Let
zn,
t:::;
ml -
1,
Paul H. Rabinowitz
334
Then
uniformly in large k . The cost as measured by
i:
L(qk)dt of qk going from
at t = -00 to 8B r (O for some b E (-00, md exceeds some "( = "((r) > O. Suppose qk does not satify
~
(3.6) Observing that 1(Pk) < 1(qk) via (3.5) and Pk satisfies (3.4) via (3.5). Thus qk could be replaced by Pk giving us a new minimizing sequence with (3.6) now being satisfied by Pk . An identical analysis proves that
(3.7) as p -+ 0 uniformly in large k and (3.8) (3.9) The L~c convergence of qk to Qm implies Qm satisfies (3.6) and (3.8)-(3.9). Moreover since r < ~, (3.6) and (3.9) imply Qm( -00) = ~ and Qm(oo) = 'f/. Hence Qm E X m .
Proof of (B) : It suffices to find q* E Xm such that
1(q*) ::;
c(~, 'f/)
+c.
Let Ql, Q2 be such that Ql E r(~,(), 1(Qd = c(~,(), Q2 E r((,'f/), and 1(Q2) = c((,'f/) . These functions are not unique. Indeed TjQi satisfies the same conditions as Qi for all j E Z and i = 1,2. Normalize Ql as follows : From {TjQl I j E Z}, choose Qi = Til Ql so that Qi(t) E C\ for all t ::; ml and Qi(t) rf. O{ for some t E (ml ,ml + IJ . Similarly from {TjQ 2 I j E Z}, choose Q2(t) so that Q2(t) E O( for t::; m3 and Q2(t) rf. O( for some t E (m3 ,m3 + IJ . For m2 = m2(p) sufficiently large, Qi(t) E O( for t ~ m2. Let
'P(t)
= (t -
m2)( + (m2
+ 1- t)Qr(m2) .
A new proof of a. theorem of Strobel
335
+ 2 and set
Let m3 > m2
1/J(t) = (t - (m3 - 1))Q;(m2)
for m4 = m4(p) sufficiently large, Q2(t) E by gluing to to (I::~~ for
Qil:: cpl::+1
+ (m3
01/ for t 2:
- t)(.
m4. Let q*(t) be defined
1/J1::_I to Q21:
3
•
Then as in (3.8),
I(q*):::; c(~,() +o(p) +o(p) +c((,1J):::; c(~,1J) +€
(3.10)
provided that p = p(€) is small enough. Step (C): Let
r*(~, () = {q E WI~'c2(1R, IRn) 1 q( -00) = ~, q(O) E 8(O£, U Oc;), q(oo) = (} and similarly r*((, 1J) = {q E WI~';(IR, IRn) 1 q( -00) = (, q(O) E 8(0( U 01/)' q(oo) = 1J} . Set c*(~, () =
inf
I(q)
inf
I(q).
qEr*(£,,()
(3.11)
and c*((,1J) =
qEr*«,1/)
Above arguments as in (A) show there exist functions PI E r*(~, () such that I(Pd = c*(~,() and P2 E r*((,1J) such that I(P2 ) = c*((,1J). Furthermore c*(~,() > c(~,() and c*(('1J) > c((,1J) . Indeed if e.g. c*(~,1J) = c(~,1J), then since PI E r(c () and [(PI) = c(~, (), PI is a solution of (HS) . Hence PI (0) E S(~,(). But PI (0) E 8(O£,UOc;) and by construction 8(O£,UOc;)nS(~,() = ¢. Consequently c* (~, 1J) > c( ~, 1J). Choose € so that O<€< The choice of
€
~min(c*(~,()-c(~,(),
c*((,TJ)-c((,1J)).
(3.12)
determines p and therefore a lower bound for m2 - ml and
m4 - m3·
Step (D): The difference m3 - m2 is still free. Suppose that Qm(mi) E 8(O£, U O( U O( U 01/) for some i. The argument proceeds in the same fashion for all cases so for convience suppose that i = l. By (3.2) it can be assumed that ml = O. By (3.8), IQm(t) -(I :::; r for all t E [m2, m3] independently of m3 -m2. Let a < < r . We claim for m3 - m2 sufficiently large, there is a subinterval, [a, a + 2] of [m2, m3] with a E N such that
IQm(t) - (I < a,
t E [a,a
+ 2].
(3.13)
Paul H. Rabinowitz
336
To see this, note first that there is some point [ E [m2, m3) such that a
IQm([) - (I <
2".
Otherwise and therefore
>
[(Qm)
{ma _ V(t, Qm)dt
(3.14)
1m2
> where f3 was defined in (2.2). Since (m3 -m2) can be made arbitrarily large, (3.14) is not possible. Indeed by (B), (3.15) for all large £ where
m2 - mI, m4 -
m2. Hence (3.15) shows on any interval of length
there exists such a point [. Now if (3.13) does not hold, in every interval of the form [a, a+2) C [m2, m3) with a E N and which contains a [ as above, there is a second point ;;y with IQm(;;Y) - (12': a. It can be assumed that IQm(t) - (I 2': ~ in [;;y,[) (or b,;;Y) as the case may be). Hence as in (2.3), (3.16) where N is the number of such intervals. But as (3.16) is not possible. Thus (3.13) holds.
m3 - m2
-+
00,
N -+
00
Step (E) : Define functions P and Q as follows P(t)
Q(t)
= =
Qm(t),
t
~
a
(t - a)( + (a + 1- t)Qm(a),
(,
t2':a+1
(,
t ~ a
a ~t~a +1
+1
(t - (a + l))Qm(a + 2) + (a + 2 - t)(, a + 1 ~ t ~ a + 2 Qm(t), t 2': a + 2.
so
A new proof of a theorem of Strobel
337
Then P E r*(~,() and Q E r((, 7]) so by (3.11) and (3.12)
I(P) + I(Q) On the other hand, by (B),
c(~, 7]) + €
> I(Qm)
~
+ 4€ + c((, 7]) c(~, 7]) + 4€. c(~, ()
>
i:
I(P) + I(Q) -
L(Qm)dt +
l:2
L(Qm)dt
(3.17)
(3 .18)
",+1 1"'+2 L(P)dt L(Q)dt. 1'" ",+1
Hence by (3.17)-(3.18),
",+1 1'"
L(P)dt +
10+2
L(Q)dt ~ 3€.
0+1
(3.19)
But by (3.14) and the definition of P and Q,
1"'+2
0+1 L(P)dt + L(Q)dt ~ x(a) (3.20) '" ",+1 where x(a) -+ 0 as a -+ O. In particular for a « r, x(a) < 3€ so (3.19) is
1
impossible. The proof of Theorem 3.5 is complete.
References 1. K. Strobel, Multibump Solutions for a Class of Periodic Hamiltoniansystems, University of Wisconsin thesis (1994) . 2. S. V. Bolotin, Existence of homo clinic motions, Vestnik Moskov. Univ. Ser. I. , Matem. Mekh. 6 (1983), 98- 103. 3. V. V. Kozlov, Calculus of variations in the large and classical mechanics, Russ. Math. Surveys 40:2 (1985), 37-71. 4. P. H. Rabinowitz, A variational approach to heteroclinic orbits for a class of Hamiltonian systems, in Frontiers in Pure and Applied Mathematics, ed. R. Dautray (1991), 267- 278. 5. E. Sere, Existence of infinitely many homo clinic orbits in Hamiltonian systems, Math . Z. 209 (1992), 27-42. 6. V. Coti Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc. 4 (1992), 693-727.
338
Paul H. Rabinowitz
7. P. H. Rabinowitz, Homoclinic and heteroclinic orbits for a class of Hamiltonian systems, Calc. of Var. and PDE 1 (1993), 1-36. 8. M. L. Bertotti and P. Montecchiari, Connecting orbits for some classes of almost periodic Lagrangian systems, J. Diff. Eq. 145 (1998), 453--468. 9. F. Alesseo, M. 1. Bertotti and P. Montecchiari, Multibump solutions to possibly degenerate equilibria for almost periodic Lagrangian systems, Z. Angew. Math . Phys ., to appear. 10. V. Coti Zelati and P. H. Rabinowitz, Multichain type solutions for a class of Hamiltonian systems, Electronic J. of Diff. Eq., to appear. 11. M. Calanchi and E. Serra, Homoclinic solutions to periodic motions in a class of reversible equations, Calc . of Var. and PDE, to appear. 12. P. H. Rabinowitz. Connecting orbits for a reversible Hamiltonian system. Dyn. Sys . and Ergodic Th .. to appear. 13. P. H. Rabinowitz, On a theorem of Strobel, in progress. 14. P. H. Rabinowitz, Periodic and heteroclinic orbits for a periodic Hamiltonian system, Ann. I.H.P.-Analyse nonlin. 6 (1989), 331-346.
Global Injectivity and Asymptotic Stability via Minimax Method Elves A. B. Silva
1
Departamento de Matematica, Univ. de Brasilia, 70910-900, Brasilia, DF, Brazil, Marco A. Teixeira 2 Departamento de Matematica, Univ. Estadual de Campinas 13081-970, Campinas, SP, Brazil. (Dedicated to Paul Rabinowitz: teacher, colleague, on the occasion of his 60 th birthday)
Abstract A new version of the mountain pass theorem is used to study the global injectivity of vector fields and the global asymptotic stability of autonomous dynamical systems on euclidean spaces. The two dimensional dynamical system is considered under hypotheses which do not imply the Markus-Yamabe condition. For dimensions greater than two, it is used the technique developed by Olech for the two dimensional problem and the direct method of Liapunov. A Palais-Smale type condition is used to describe the behavior of the unbounded orbits.
1
Introd uction
Since the seminal work of Ambrosetti-Rabinowitz [1], the minimax theory has been widelly applied to the problems of existence and multiplicity of nonlinear differential equations [20, 24, 2,13). One of the major results in [1) is the famous mountain pass theorem. Without any doubt, this theorem has been the main motivation for the development and application of the minimax theory for the last three decades. Our intention in this work is to present applications of the minimax method to the study of global univalence of vector fields on euclidean spaces and to the problem of global asymptotic stability for autonomous dynamical systems. The argument used here are based on the the method developed in [22, 23). In order to apply the minimax method to the problems considered, we use a version of the mountain pass theorem, proved by the authors [22), that relates the topology of the level surface of a functional of class C 1 defined in a real Banach space with the existence of critical points for the functional. To be able to state this last result, we need to introduce some preliminary informations. lSupported by CNPq/Brazil and Pronex: " Equ~6es Diferenciais Parciais nao Lineares" . 2Supported by CNPq/Brazil, FAPESP /Brazil and Pronex: "Teoria Qualitativa das Equ~6es Diferenciais Ordinarias".
339
E . A. B. Silva and M. A . Teixeira
340
Let E be a real Banach space. Given fECI (E, JR) and c E JR, we denote by Sc(j) and Kc the sets {u E E : f(u) = c} and {u E E : f(u) = c, f'(u) = O}, respectively. We say that c is an admissible level of f if either c is a regular value of f or the components of Kc possess only a point and c is an isolated critical value of f. We now recall the compactness condition introduced by Palais-Smale [1, 20]:
Definition Given fECI (E, JR) and c E JR, we say that f satisfies the PalaisSmale condition at level c E JR, denoted (PS)c , if every sequence (Uk) C E satisfying f(Uk) --+ c and IIt(Uk)11 --+ 0, as k --+ 00, possesses a converging subsequence. If f satisfies (PS)c for every c E JR, we just say that it satisfies (PS).
Our main result is the following version of the mountain pass theorem,
Theorem A (The Level Surface Theorem) Suppose fECI (E, JR) satisfies (PS) . Assume that c E JR is an admissible level of f and that U and v are two distinct points of Sc(j) . Then, either (i) U and v are in the same path-component of Sc(j), or (ii) f has a critical value d f:. c.
Note that the Theorem A is a generalization of the mountain pass theorem since the hypothesis of this last result implies that So(j) have two distinct and nonempty path-components. As in the mountain pass theorem, we establish a minimax characterization of the possible values of d. The level surface theorem can also be seen as a version of Rolle's Theorem for functionals defined in real Banach spaces. For the sake of completeness, we present a proof of Theorem A in section 2. In our first application of the level surface theorem, we consider the problem of global injectivity for vector fields defined in the two dimensional euclidean space. We note that results on global univalence for vector fields have been the object of intensive research in recent years. In several branches of the Mathematics important questions are related to such results [18, 15, 9, 19]. Given c E JR2 and X : JR2 --+ JR2 of class Cl, we follow the notation used above and we set Sc(X) = X-l({C}). We denote by K(X) the set of singular points of X, and we represent by Kc(X) the set K(X) n Sc(X). Writing X = (lI,h), we suppose (HI) JR \
II (K(X))
is a dense subset of JR. (H2) \l1I(u) f:. 0, for every u E JR 2. As a consequence of Theorem A, we establish:
Theorem B ( A Global Injectivity Theorem) Suppose X = (II, h) E C 1 (JR 2, JR2) satisfies (Hl) -(H2), with II satisfying (PS) . Then, X is globally injective provided Sc(X)
n (JR2 \
K(X))
f:. 0 whenever Sc(X) f:. 0.
Global injectivity and asymptotic stability
341
Theorem B is an improved version of the global injecitvity theorem proved in [22] since in that article it is supposed that X is a local diffeomorphism. Note that the above result is independent of the choice of the coordinates system, and that the (PS) condition is assumed for just one coordinate of X. We should also mention that in [22] it is also considered the problem of global injectivity on higher dimensions. To derive such results, a deformation lemma based on the minor determinants of the Jacobian matrix of the vector field is proved. Finally, we observe that Theorems A and B also hold under a generalized versions of the (PS) condition (see Remarks 2.6 and 3.3). In the second part of this article, we show how the technique developed by the minimax method can be used to study the global asymptotic stability for the autonomous dynamical system (AS)
u(t) = X(u(t)),
where X : mm -+ mm is a vector field of class 0 1 satisfying X (0) = O. We also suppose that the eigenvalues of X'(O) have negative real parts. By Xm , we denote the family of vector fields on mm satisfying those assumptions. Note that the origin is an asymptotic attractor for the system (AS) whenever XEXm . For the last four decades, one of the main reasons for the study of the global asymptotic stability for the system (AS) has been the conjecture stated by Markus-Yamabe in [12] : Conjecture (Markus-Yamabe) Suppose X E Xm satisfies (MY) The eigenvalues of X' (u) have negative real parts for every u E mm. Then, the system (AS) is globally asymptotically stable.
The answer to the above conjecture depends on the dimension of the euclidean space mm. A key stone for the proof of the conjecture on the two dimensional euclidean space was established by Olech [17] in 1963. Based on this famous work, in 1993 the Markus-Yamabe Conjecture was proved for m = 2 by Gutierrez [9], Fesler [7] and Glutsyuk [8]. We note that in 1987, Meisters-Olech [15] proved the conjecture on m2 for polynomial vector fields. The Markus-Yamabe Conjecture was completely solved in 1995 when a polynomial counter-example was found on m3 by Cima et al [4] while those authors where studying the famous Jacobian Conjecture. Since its announcement in 1960, the Markus-Yamabe Conjecture and related problems have been intensively studied. There exists a vast literature on this subject. We refer the interested reader to [15, 16, 9, 7, 8, 4] and references therein. In particular, we cite [14] where the history of the Markus-Yamabe Conjecture and its relation to the Jacobian Conjecture are presented. The results proved in this article for the problem (AS) are also based on
E. A. B. Silva and M. A. Teixeira
342
the dimension of the euclidean space JR m . First, we state a theorem on the global asymptotic stability for the two dimensional problem when the (MY) condition does not hold in all space. More specifically, we suppose (H2), the following version of (HI) (H3) There exists c>
°such that (-c, c) \
ft (K(X)) is dense on (-c, c),
and (H4) Trace(X'(u)) < 0, for every u E JR 2. As a consequence of the level surface theorem and Olech's result (17), we have
= (ft, h) E X 2 satisfies (H2)-(H4), with ft satisfying (PS). Then, the system (AS) is globally asymptotically stable.
Theorem C Suppose X
Observe that conditions (H2)-(H4) provide a weaker version of the (MY) condition for the two dimensional problem since, on this case, this last hypothesis implies that DetX'(u) > and TraceX'(u) < 0, for every u E JR 2 . In (23), it is assumed a stronger version of (H3). The argument employed here for the systems in more than two dimensions combines the direct method of Liapunov (21) with the method used by Olech (17). Given m ~ 3, we write JRm = JR2+n = JR2 X JR n , and we suppose that the plane JR2 is an invariant set on which the hypothesis of Theorem B is satisfied. Supposing the existence of a Liapunov function on JRm \ JR2 satisfying a Palais-Smale type condition with respect to the vector field X, we are able to show that the invariant two dimensional plane is a global attractor for the system (AS). Then, assuming two technical conditions, we verify that the origin is a global attractor for (AS). Since we suppose that the plane JR2 is an invariant set, we may write X = (L + H,G) with X((x,O)) = (L(x),O), for every x E JR 2 . Henceforth, we denote by Xm , m ~ 3, the family formed by the vector fields X = (L + H, G) defined on JRm which have the plane JR2 as an invariant set with L satisfying the hypothesis of Theorem C.
°
As observed above, our results for the systems in more than two dimensions are also based on the existence of a Liapunov function for the system (AS) . More specifically, we suppose (H5) There exists a function V E C l (JR 2+n , [0, 00)) satisfying (i) inf{V(u) : u
= (x,y), Ilyll ~ 6} > 0, for every 6 > 0,
(ii) (\7V(u),X(u)) < 0, for every u E JR2+n \ JR 2. It is worthwhile mentioning that condition (H5) does not imply that the origin is a global attractor for (AS). To be able to show that the plane JR2 is a
Global injectivity and asymptotic stability
343
global attractor for the system (AS), we introduce a version of the Palais-Smale condition for a functional with respect to a given vector field X:
Definition Given X E C (mm , mm), we say that V E C 1 (mm ,m) satisfies the (PS) condition with respect to X at level c E m, denoted (PS)(X,c), if every sequence (Uk) c mm such that V(Uk) -t c and (\7V(Uk), X(Uk)) -t 0, as k -t 00, possesses a convergent subsequence. Note that V E C 1 (mm, m) satisfies the Palais-Smale condition at level cE if it satisfies (PS)(,vv,c). Throughout this paper we let M denote the set x {O}. Observing that a solution of (AS) satisfies the (positive) semicomplete condition if it is defined on [0,00), in [23) it is proved the following basic result:
m m2
Lemma A ( The Fundamental Lemma) Suppose X E Xm , m ~ 3, satisfies (H5) with V satisfying (PS)(X,c) for every c > 0. Assume further the semicomplete condition for the solutions of (AS) . Then, the plane M is a global attractor for the system (AS). Lemma A was motivated by the observation that a version of the PalaisSmale condition may be combined with the direct method of Liapunov to study the behavior of the orbits of a dynamical system. Since the plane M is a global attractor for any semi-complete solution of the counter-example to the Markus-Yamabe Conjecture [4), we do not expect that the origin is a global attractor for (AS) under the hypothesis of Lemma A. To overcome such difficulty, we assume the following two technical conditions: (H6) There exist c, M, R > IIxil > R, Ilyll < p, we have
°
and p E (0,00) such that, for every u = (x, y),
(i) I(L(x).l ,H(u))1 ~ MV(u), (ii) (\7V(u),X(u))
~
-cV(u),
and (H7) There exists 8 E [0,1) such that lim IIH(x,y)II IIxll-+oo,lIyll-+o IIL(x) II
< 8.
-
In (H6), L.l represents the vector field orthogonal to L, obtained by a counterclockwise rotation. Now, we may state:
Theorem D Suppose X E Xm satisfies (H5)-(H7), with V satisfying (PS)(X,c) for every c > 0. Assume further the semi-complete condition for the solutions of (AS). Then, the origin is a global attractor for the system (AS) .
E. A. B. Silva and M. A. Teixeira
344
We should mention that Theorem D was strongly motivated by the counterexample to Markus-Yamabe Conjecture on m3. In our setting, the technical conditions (H6)-(H7) provide the necessary estimates for us to apply a version of Olech's argument, showing that the orbits converging to the plane should follow the flow of the orbits on the invariant plane. For the sake of completeness, we present the proofs of Theorem A and Lemma A in sections 2 and 4, respectively. In section 4, we also state two technical results that give us some estimates for the length of a curve on m2 which has the origin on a bounded component of its complement. Those estimates are used in the proof of Theorem D. In section 3, we present the proofs of Theorems Band C. In section 5, we give an outline of the proof of Theorem D. Finally, we reserve section 6 to present our final remarks on the the results considered in this article. There, we also recall a conjecture presented in [23) which is related to a recent result by Gutierrez-Teixeira [10).
2
Proof of Theorem A
In this section we present a proof of Theorem A [22). First. we recall a deformation lemma due to Chang [2), Proposition 2.1 (A Deformation Lemma) Suppose f E C1(E, m) satisfies (PS). Assume that a is the only possible critical value of f on the interval [a, b) and that a is an admissible level. Then, there exists a continuous map r : [0,1) x (fb \ Kb) -+ fb \ K b, so that (i) r(O, u) = u, VuE fb \ Kb (ii) r(t, u) = u, V (t, u) E [0,1) x (iii) f(r(l,u)) = a, VuE fb \ (Kb u r)
r
Before proving Theorem A, we also need to establish two preliminary results. Considering u, v E Sc(f), given in the hypothesis of Theorem A, we define Cl
= '"fEr, inf max f(r(t)), tE[O,lj
(2.1)
with
r 1 = hE C([O, 1), E) I ,(0) = u,
,(1)
= v}.
(2.2)
As a consequence of Proposition 2.1, we have
Lemma 2.2 Suppose c is an admissible level of f. Then. either Cl is a critical value of f, or there exists, E r 1 such that max f(r(t))
tE[O,lj
< c. -
>C
and Cl
(2.3)
Global injectivity and asymptotic stability
345
Proof: It is clear that c ~ Cl < 00. Moreover, when c < CI, Proposition 2.1 and a standard minimax argument imply that Cl is a critical value of f. Hence, we may suppose that Cl = c. Since c is an admissible level of f, there exists E > 0 such that K n f-l([C - E, C + ED = Kc . Furthermore, by (2 .1), we have 1'1 E r 1 such that max f(')'l(t)) < C+E.
tE[O,l)
-
Invoking Proposition 2.1 one more time, we obtain 7(t, u) satisfying (i)-(iii) with a = C and b = C + E. It is not difficult to show that 1'(t) = 7(1,1'1 (t)) belongs to r 1 and satisfies (2.3) . The lemma is proved. 0 We now define C2
= sup min f(')'(t)),
(2.4)
1'Er2 tE[O,l)
with
r 2 = bE C([O, 1], r) 11'(0) = u,
1'(1)
= v},
(2.5)
to obtain Lemma 2.3 Suppose that C is an admissible level of f and that r 2 =f 0. Then, either C2 < C and C2 is a critical value of f, or there exists l' E r 2 such that
f(')'(t)) = c, 'If t E [0,1].
(2.6)
Proof: We have that -00 < C2 ~ c. As in Lemma 2.2, C2 is a critical value of C2 < c. For C2 = C, we apply Proposition 2.1 for - f with a = -c , b = -c + E, and E > 0 sufficiently small. Then, arguing as in. the proof of Lemma 2.2, we find a path l' E r 2 satisfying (2.6) . Lemma 2.3 is proved. 0 Proof of Theorem A: Suppose that u and v are not in the same pathcomponent of Sc(f). Considering Cl and C2 given by (2.1) and (2.4), respectively, we claim that at least one of those numbers is a critical value of f. Effectively, if we suppose otherwise, Lemma 2.2 implies that r 2 =f 0. Consequently, by Lemma 2.3, we must have l' : [0,1] -+ Sc(f) such that 1'(0) = u and 1'(1) = v. But, that contradicts the fact that u and v are not in the same path-component of Sc(f). On the other hand, if f has no critical value d =f c, then, necessarily, Cl = C2 = C and ~ and v are in the same path-component of Sc(f). That concludes the proof of Theorem A. 0
f whenever
Remark 2.4 {i} It is clear from the proof that Theorem A holds if we assume (PS)c for C E [Cl' C2], where Cl and C2 are given by {2.1} and {2.4}, respectively. {ii} Replacing f by -fin {2.1}-{2.2} and {2.4}-{2.5}, we obtain two other possible critical values for the functional f.
E. A . B. Silva and M. A. Teixeira
346
We now state a natural generalization of Theorem A that was also proved in [22]. Given a topological space X, we denote by Hk(X) the k-th reduced singular homology with integer coefficients. Theorem 2.5 Suppose f E CI(E,JR) . Assume that c E JR is an admissible level of f and that H.(Sc(f)) is not trivial. Then f possesses a critical value d::fi c. Remark 2.6 In [22} it is proved that the Proposition 2.1 is true under a generalized version of the (PS) condition. Hence , we may conclude that Theorem A holds when that hypothesis is supposed.
3
Proofs of Theorems Band C
The following consequence of the level surface theorem provides the proofs of the Theorems Band C, Lemma 3.1 Suppose that X = (h,h) E C I (JR 2,JR2) satisfies X(O) = 0, Det(X'(O)) ::fi 0 and (H2)-(H3), with h satisfying (PS). Then, X satisfies (H8) X(u) ::fi 0, for every u E JR2 \ {O}, (H9) There exist P, a
> 0 such that
IIX(u)11
~
a, VuE
JR2,
Ilull
~ p.
Proof: Invoking the inverse function theorem, we obtain two open balls centered at the origin, BPi (0) C JR2, i = 1,2, such that X : BpI (0) --t JR2 is injective and B p2 (O) C X(Bpl (0)). Thus, to prove Lemma 3.1, it suffices to show that (H9) holds with P > PI and a = min{P2, c}, with c given by (H3). Arguing by contradiction, we suppose that there exists u E JR2 such that IIX(u)11 < a and Ilull ~ Pl · By our choice of a, we have that Ih(u)1 < c and X(u) E B(O,P2) . Furthermore, by (H2)and (H3) and we may suppose that CI = h(u) rf. h(K(X)) Now, let v E BpI (0) be such that X(v) = X(u). Since h satisfies (PS), by (H2) and Theorem A, there exists "( : [0,1] --t JR2 such that "(0) = v, "(I) = u, and hb(t))
= CI = h(u),
V t E [0,1].
(3.7)
Considering h : [0,1] --t JR2 defined by h(t) = hb(t)), for t E [0,1], we have that h(O) = h(l) = h(u) . Furthermore, from (H2), (3.7) and the implicit function theorem, we find to E (0,1) such that h'(to) =< hb(to)), "('(to) >= 0,
Global injectivity and asymptotic stability
347
< ft(-y(to»,')"(to) >= 0, and ,),'(to)::J. O. This implies that Det(X'(-y(to») = 0, contradicting our choice of Cl. The proof of Lemma 3.1 is concluded. 0 For the proof of Theorem B, given Sc(X) ::J. 0, we take Uo E Sc(X) \ K(X) and apply Lemma 3.1 to the vector field F(u) = X(u + uo) - c, obtaining that X(u) ::J. c for every u E m? \ {uo} . 0 Theorem C is a direct consequence of Lemma 3.1 and Olech's result:
Lemma 3.2 Suppose X E X 2 satisfies (H4), (H8) and (H9). Then, the system (AS) is globally asymptotically asymptotically stable.
Remark 3.3 (i) As observed in the introduction Lemma 3.2 is the key stone for the proof of the two dimensional Markus- Yamabe Conjecture. (ii) We also note that Theorems Band C are also true under the generalized version 0 the (PS) condition assumed in [23}.
4
Proof of Lemma A and Preliminary results
Given Uo E m2+n, we will show that Uo is attracted by M. By Theorem C and the fact that X E Xm , it suffices to consider Uo fj. with ')'(t) = ')'(t, uo) satisfying
m2
11')'(t)11 fi 0,
as t -+ 00.
(4.8)
Following the standard notation for Liapunov functions, we set V(t) { V(t)
= V(-y(t», = ~~ (t) = (\7V(-y(t»,X(-y(t»).
The next lemma is a consequence of (H5) and the fact that the origin is an asymptotic attractor for the system (AS) .
Lemma 4.1 Suppose X E Xm satisfies (H5). Assume,), = ')'(. ,uo) : [0,00) -+ (4 .8). Then, Ib(t)II -+ 00 as t -+ 00.
m2+n is a solution of (AS) satisfying
We now conclude the proof of Lemma A. As observed before, it suffices to consider,), = ')'(.,uo) satisfying (4.8). We claim that there exists a sequence tk -+ 00, as k -+ 00, such that
V(tk) -+ 0, as k -+ 00. Effectively, if we assume otherwise, we find T > 0 and K > 0 such that V(t) ::; -K, for every t ~ T. But this implies V(t) -+ -00 , as t -+ 00,
E. A. B. Silva and M. A . Teixeira
348
contradicting (H5). The claim is proved. Next, invoking Lemma 4.1 and the fact that V satisfies (PS)(X,c), for every c > 0, we conclude that V(tk) -+ 0, as k -+ 00. Therefore, V(t) -+ 0, as t -+ 00, since 0 < V(s) ::; V(t), for every s ~ t. By (H5), we finally get that Ily(t)11 -+ 0, as t -+ 00. 0 As a consequence of the proof of Lemma A, we have Corollary 4.2 Suppose X E Xm , m ~ 3, satisfies (H5). Assume 'Y(., uo) = (x(.),y(.)) : [0,00) -+ JR2+ n is a solution of (AS) satisfying (4.8). Then, Ilx(t)11 -+ 00 and Ily(t)1I -+ 0, as t -+ 00 . The next results provide estimates for the arclengths of curves on JR2 \ {O} that have the origin belonging to a bounded component of their complement. Given a continuous curve (3 : [0, 1] -+ JR2, we denote by 1((3) = 1((3([0, 1])) its length. Lemma 4.3 Suppose (3 : [0, 1] -+ JR2 is a continuous closed curve such that
((3d the origin belongs to a bounded component of JR2 \ (3([0, 1]), ((32) there exist to E [0,1] and d > 0 such that
11(3(to)11 ~ d > O. Then. 1((3)
~
2d.
Proof: Without loss of generality, we may suppose that to = o. By ((3d, there exist t E (0,1) and oX > 0 such that (3(t) = -oX(3(O). Consequently, by ((32), 1((3) = 1((3([0. t]) + l((3([t, 1]) ~ 2d. The lemma is proved. 0 Corollary 4.4 Let (3 : [0,1] -+ JR2 \ {O} be a piecewise C 1 simple closed curve satisfying ((32). Suppose T : JR2 -+ JR2 is a vector field of class Cl satisfying (Td T(O) = 0 and T(x) ::p 0, for every x E JR2 \ {O}, (T2) (T((3(t)), ((3'(t)).l) ~ 0(::; 0), for every t E [0,1] such that (3'(t) is defined. Then, 1((3)
~
2d.
Remark 4.5 Note that Lemma 3.1 implies that L, L.l : JR2 -+ JR2 satisfies (T1) whenever L satisfies the hypothesis of Theorem C.
Global injectivity and asymptotic stability
5
349
Proof of Theorem D
To prove Theorem D, we argue by contradiction and suppose that (AS) possesses a solution ')'(t) = (x(t),y(t)) = ')'(t,uo) satisfying (4.8) . As observed before, we have that ')'(t , uo) ¢ M , for every t 2:: O. In the following, we set F=L+H. Our main argument is a variation of Olech's method for the two dimensional problem. To use this argument, we consider x(t), the projection on M of the solution ')'(t,uo), and ')'(t,xo), the solution ')'(t , xo) of (AS) with the initial condition Xo E M. Then, Ilx(t)11 -+ 00 and Ih(t,xo)11 -+ 0, as t -+ 00, by Corollary 4.2 and Theorem C, respectively. Next, given T E m, we consider 1/(t,T) = 1/(t,')'(T,XO)), the solution in M of the system x(t) = L.L(x(t)), { x(O) = ')'(T,Xo) E M. Suppose the existence of T = T(T) and S = S(T) 2:: 0, T 2:: 0, such that x(S) = 1/(T, T) and T(T) -+ 00, as T -+ 00. If we know that
r = ')'([0, T)) U 1/«0, TJ, T) U x«O, S)) is a simple closed curve, we may apply Green's Theorem on the bounded component of m2 \ r to estimate the length of 1/([0, TJ, T). For this, we use an estimate for the function Ro(t)
= foo I(L(x(s)), F.L(x(s), y(s)))1 ds,
which we call the rate of the flow of Lacross x([O, 00)). That argument would provide a contradiction to Ilx(t)11 -+ 00 and ')'(T, xo) -+ 0, exactly as in Olech's argument [17] . However, we note that the claim that r is a simple closed curve is not true in our setting. Furthermore, we do not have T = T(T), S = S(T) well defined in general. To overcome these difficulties, given s, T > 0, we find sequences TO = 0 < T1 < .. . < Tj :::; T and So = 0 < Sl < ... < Sj :::; s such that, for every T E h-1,Tj), there exist unique T(T) and S(T) E [Sj-1,Sj) so that X(S(T)) = .,,(T(T), T) and l(1/([O, T(T)], T) < R. Moreover,
is a simple closed curve. Applying Green's Theorem on the bounded component of m2 \ r T, with T -+ Tj, after a finite number of steps, we verify that either Tj = T, or Sj = s ~ Then, taking T and s sufficiently large, we show that this contradicts the estimate for the length of 1/([0, T(T)J, T) .
E. A. B. Silva and M. A. Teixeira
350
Since the argument employed here is similar to the one used in [23], we just provide an outline of the proof of Theorem D, proving some of the main steps. Considering R, p'> 0 given by (H6), taking R > 0 larger and p > 0 smaller if necessary, we invoke the Lemma 3.1. X E Xm and (H7) to find d > 0 and o ~ 8 < 1 such that
IIL(x)11 ~ d > 0, { IIH(x,y)11 ~ 81IL(x)ll,
'if Ilxll ~ R, 'if Ilxll ~ R, Ilyll ~ p. Then, applying Corollary 4.2, we find T ~ 0 such that
IIx(s)11 ~ 3R, { lIy(s)11 ~ p,
'if s ~ T, 'if s ~ T. The following lemma provides an estimate for Ro(t) . Lemma 5.1 There exists Tl
Ro(T1 )
=
1
00
~ T, T
(5.9)
(5.10)
given by (5.10), such that
I{L(x(s)) , Fl.(x(s), y(s)))1 ds <
~
T'
T,
Proof: By X E Xm , the statement (i) of hypothesis (H6) and (5.10), for every S ~ T, we get
fsoo I{L(x(s)),Fl.(x(s),y(s)))lds ~ b fsoo V(s)ds. On the other hand, by the statement (ii) of hypothesis (H6) and (5.10), we have V(s) ~ V(S)e-c(s-S). for every s ~ S ~ T . Hence,
roo V(s) ds ~ bV(S) roo e-c(s-S) ds = bV(S).
is
is
c
Since V(S) -+ 0, as S -+ 00, we obtain the desired estimate by taking Tl = S > 0 sufficiently large. The lemma is proved. 0 Remark 5.2 It is worthwhile mentioning that (H6) is used only to prove the Lemma 5.1 . Thus, any other condition that establishes this result provides the global asymptotic stability for the system (AS) (see examples in [23]).
Considering XT, = (x(Td,O) EM. we take ,(t, XT,), the solution of (AS) with ,(0) = XT,. Since X E Xm , we have that ,(T,XT,) E M , for every T E m, and ,(T, XT,) -+ 0, as T -+ 00. Furthermore, using that the system (AS) is autonomous, it is not difficult to show that we may suppose Tl = T = 0 and
X«O,oo)) n,«O,oo),xo) = 0. As a direct consequence of (5.9) and (5 .10), we have
(5.11)
Global injectivity and asymptotic stability
351
Lemma 5.3 The mapping x: [0,00) --+ m2 is locally injective.
Now, given 7 E m, we denote by (W-(7),W+(7)) the maximum interval of definition for the solution 7J(t,7) = 7J(t,,(7,XO)) of (AS)l., we set
0= {(S,t,7) E m3 : s E m,7 E m,t E (W-(7),W+(7))}, and we define ~ : 0 --+ m2 by ~(s, t, 7) = x(s) - 7J(t, 7), for (s, t, 7) E O. The following Proposition is a direct consequence of (5.9), (5.10) and the implicit function theorem.
°
Proposition 5.4 Given (so, to, 70) E 0 such that ~(so, to, 70) = and So ~ 0, we may find a neighborhood Uro of 70 and unique functions of class C 1 , ¢>l (7), ¢>2( 7) : Uro --+ m such that (¢>d 70), ¢>2( 70)) = (so, to) and ~((¢>l (7),
¢>2(7), 7))
= 0,
V 7 E Ura-
If s E ¢>1(Uro), t E ¢>2(Uro) and 7 E Uro satisfy ~(S,t,7) = 0, then (s,t) = (¢>l (7), ¢>2(7)). Furthermore, when (so, to, 70) = (0,0,0), we may suppose that ¢>l : Uo --+ m is an increasing function.
Remark 5.5 By (5.11), Proposition 5.4, and the fact that ,(t, xo) is not periodic, we have ¢>2(7) '" 0, for every 7 E Uo, 7 > 0. Since the proof of the other case uses a similar argument, without loss of generality, we suppose that ¢>2(7) > 0, for every 7 E Uo , 7 > 0.
We now give some estimates for the arclengths of the mappings x(s) and > and 7' > be such that
,(7,XO)' Let M
°
°
2R < 111'(7, xo)1I ~ M, V 7 E [0,7'], { 11!(7',xo)1I = 2R.
By Corollary 4.2, there exists 5>
(5.12)
°
such that
Ilx(s)1I ~ M
+ 2R,
V s ~ 5.
(5.13)
Note that by (5.10), (5.12) and (5.13), we have
Taking MI
°
IIX(5) -,(7,xo)11 ~ R, V 7 E [o,~], { 11,(7', xo) - x(s)1I ~ R, V s E [0, s1·
> such that IIL(x(s)) + H(x(s),y(s))11 ~ M!, V { IILb(7,Xo))1I ~ M 1 , V ~ 7 ~ 7',
°
°
~ s ~ 5,
(5.14)
(5.15)
E. A . B. Silva and M. A. Teixeira
352
we set,
{
71 = min{ 2~1' f}, {R -} SI = mm 2Ml' S . A
•
From (5 .15), we obtain
l(x([O,sd» < R, { l(r([O,71J,XO» < R.
(5.16)
Such inequalities will be used in the proof of the next result.
Lemma 5.6 The mapping x : [0, sd
-t
JR2 is injective.
Proof: Arguing by contradiction, we suppose that there exist 0 :::; si < S2 :::; SI such that x(si) = x(s2). First, we claim that we may assume that x : [si, S2] -t JR2 is a simple closed curve. Effectively, considering S2
= sup{ s :::: si
: x : [si, S2]
-t
JR2 is injective},
by Lemma 5.3, we have that si < S2 :::; S2. Using the definition of S2 and Lemma 5.3 one more time, we find 0 < f < S2 - si , to E [SI, S2 - f] and sequences (tk) C [SI,S2-fJ, (Tk) C [S2,S2+f] such that x : [S2-f,S2-f]-t JR2 is injective, X(tk) = x(Td, for every k E IN, and tk -t to, Tk -t S2, as k -t 00. Hence, x(to) = X(S2) . Moreover, it is not difficult to verify that x : [to, S2] -t JR2 is a simple closed curve. The claim is proved. From (5.10), we derive that x[si, S2] -t JR2 satisfies (fh), with d = 3R, and x([si, sm C JR2 \ {O}. By (5.9), (5.10) and X E Xm , L1. is transversal to x([si,sm. Hence, L1. satisfies (T2). Since L1. satisfies (Tl ), we may invoke Corollary 4.4 to conclude that l(x([si, S2])) :::: 6R. But, this contradicts (5.16) . The lemma is proved. 0 Now, we consider Al C [0,71), the set formed by the points that there exist t E [O,W+(T» and s E [O,sd satisfying
77(t, T) = x(s), { l(77([O, t]), T) < R.
T
E [0,71) such
(5.17)
Note that Al -# 0 because 0 E AI. Moreover, considering [O,7d with the topology induced by the real line, by Proposition 5.4 and an argument similar to the one employed in the proof of Lemma 5.6, we obtain
Lemma 5.7 Al is an open subset of(O, 7d . Furthermore, given TEAl, there exist a unique t E [0, W+(T» and a unique s E [0, sd satisfying (5.17).
Global injectivity and asymptotic stability
353
Based on Lemma 5.7, we may define Tl : Al -t [0, (0), SI : Al -t [0, sI) by Tl (T) = t, SI(T) = s, where t, s are given by (5.17). Taking VI = [O,Tl) c [0,1'1)' the component of Al which contains the origin, we have
Lemma 5.8 Tl , SI : [0, TI) -t 0, S1(0) = 0 and
m are continuous functions
satisfying T1 (0)
=
(pd T1(T) > 0,0 < SI(T) < S1, for every T E (O,Td, (P2) There exists M > 0 such that
(P3) SI : [0, TI) -t
m is an increasing function .
Proof: By definition, T1 (0) = 0 and S1 (0) = O. Furthermore, an argument based on Proposition 5.4 shows that T l ,SI : [O,TI) -t m are continuous and satisfy (pd . Now, from (5.12) and (5.17), we get R ~ 1117(t, T)II ~ M
+ R, V t E [0, T1 (T)], T E [0, T1)'
(5 .18)
Therefore, by [9] and (AS)1., we find 8 > 0 such that 117j(t, T)II 2: 8 > 0, for every t E [0, T1 (T)]. Consequently, invoking (5.17) one more time, we obtain
Hence, (P2) holds. Finally, we shall verify (P3)' By Proposition 5.4 and Lemma 5.6, we get S1(T) = ¢>t(T), for every T E Uo, and 8(0) > O. Consequently, S1 is locally injective and increasing on a neighborhood of T = O. Hence, to prove (P3) it suffices to verify that S1 is locally injective on (0, TI)' Arguing by contradiction, we suppose there exist TO E (0, Td and sequences (Tf), (Tf) C (0, TI) such that T~ -t TO, ask -t 00, i = 1,2, Tf ¥: Tf' V k E IN, { 0< S1(Tf) = SI(Tf), V k E IN.
By the transversality of £1. and 'Y«O, (0), xo), there exists
(5.19) f
> 0 such that if
t E (-f, f), T E (TO - f, TO + f) and 17(t, T) E 'Y«TO - f, TO + f), xo), then t = O. By (5.19), 'Y(Tf,xo) = 17(T1(Tf) - Tt{Tf) , Tf), for every k E IN. Using the continuity of TI : [0, Tl) -t and the first relation in (5.19), we obtain that Tf Tf for k sufficiently large. But, this contradicts Tf ¥: Tf, for every k E IN. The lemma is proved. 0
=
m
We now consider a sequence 0 < 1'1 < ... < 1'k < ... < TI ~ 1'1 satisfying 1'k -t TI, as k -t 00. We also consider 0 < S1(1'l ) < ... < S1(1'k) < ... < S1 and
E. A. B. Silva and M. A. Teixeira
354
0< TIU't}, .. .,T l (fk), . . . < 00 the associated sequences. By (P3), SI(fk) /' SI :::; 81 . Furthermore, invoking Lemma 5.8-(P2), we may suppose without loss of generality that T l (fk) --t tl, as k --t 00.
(5.20)
As a direct consequence of (5.17), we have (5.21)
Now, invoking Corollary 4.4 and using the argument employed earlier, we conclude that the curve
is a simple closed curve for every 7 E (0,71) . Then, Taking B l ,k the bounded component of JR2 \ r fk , we apply Green's Theorem to L, deriving Lemma 5.9 Considering tl given by (5.20), we have
itt
10 IIL(1J(t, 71))11 2 dt :::;
10tl (L(x(s)), F.l(x(s), y(s))) ds.
As a consequence of the above result, (5.9), (5.18) and Lemma 5.1, we havei a strict inequality on the second relation of (5.21), (5.22)
Hence, by (5.10), (5.14), (5.21) , Proposition 5.4 and Lemma 5.7, we conclude that 71 < T , SI < s and either 71 = fl or SI = 81 . Consequently, 71
+ SI 2':
R 2Ml .
(5 .23)
For the next step, we follow the same argument. Set
{:2 == m~n{71 ++ 2i ,~}, l
S2
mm{sl
2Ml 's}.
By (5.15), (AS) and (AS).l, we also have l(X([SI,82]))
(5.24)
Global injectivity and asymptotic stability
355
Moreover, x : [SI, 82] -+ m? is injective. Now, we consider A2 C h, 72), formed by the points T E [Tl,72) such that there exist t E [O,W+(T)) and s E [SI,82) satisfying 1](t, T) = x(s), { l(1]([O, t]), T) < R.
(5.25)
By (5.21) and (5.22), Tl E A 2. Furthermore, as in Lemma 5.7, A2 is an open subset of [Tl, 72) and, given T E A 2, there exist a unique T E A2 and a unique t E [O,W+(T) and s E [Sl,82) satisfying (5.25). As before, we define T2 : A2 -+ [0,00) and 8 2 : A2 -+ [Sl, 82) to be, respectively, the values t and s provided by (5.25). Taking V2 = [Tl,T2) C [Tl,72), the component of A2 which contains Tl, we also get that T 2, 8 2 : [0, T2) -+ m are continuous functions satisfying T2 (Tl) = tl, 8 2 (Tt) = SI and
(ih) T 2(T) > 0,
< 8 2(T) < 82, for every T E (Tl,T2), (fi:!) There exists M > 0 such that
(P3) 8 2
:
SI
h, T2) -+ m is an increasing function.
Now, we consider a sequence Tl < 1"1 < ... < Tk < ... < T2 ~ 72 satisfying Tk -+ T2, as k -+ 00. We also have the associated sequences (82(Tk)) C [SI,82), (T2 (Tk)) C m. Without loss of generality, we may suppose that 8 2 (Tk) /'" S2 ~ 82, T 2(Tk) -+ t2, as k -+ 00. Moreover, (5.26) Noting that
is a simple closed curve for every T E (Tl,T2), we take B 2,k the bounded component of m2 \ r Tk' By Green's Theorem and Lemma 5.9, we get
1t2
IIL(1](t, T2))11 2 dt
~
1
82
(L(x(s)), Fl.(x(s), y(s))) ds.
As a direct consequence of (5.9), (5.18), Lemma 5.1 and the above inequality, we have
E. A. B. Silva and M. A. Teixeira
356
Hence, by (5.10), (5.14), (5.26), Proposition 5.4 and Lemma 5.11, we have that < T. 82 < s and either 1'2 = 7"2 or 82 = 82 . Consequently, by (5.23) and (5.24),
1'2
1'2
+ 82 2:
1'1
R R + 81 + 2Ml 2: Ml .
Arguing in a similar way, we obtain a sequence ((tk,8k,1'k)) c tk E (0.00), 8k E (8k-l,S), Tk E (Tk-l,T), for every k E IN. and Tk
But, this contradicts T
6
+ 8k 2:
m3 , such that
kR 2M ' V k E IN . l
+ S < 00 and concludes the proof of Theorem D.
0
Final Remarks
In [19], Rabier used a deformation result and arguments of differential topology to generalize Hadamard Theorem on global diffeormophism for euclidean spaces. We notice that in [19] the author considers a notion of admissible flow which is closely related to the generalization of the Palais-Smale condition assumed in [22]. We also note that in a recent article [11], Katriel studied global homeomorphisms for certain topological and metric spaces by proving two versions of the mountain pass theorem without assuming that the functional is of class C l . It is natural to ask for the posibility of obtaining topological versions of the level surface theorem. Ad observed in the introduction, the condition (PS)(X,c) may be very useful when the direct method of Liapunov is applied to problems possessing some lack of compactness. In particular, as we shall see in a forthcoming paper, that condition provide a natural generalization of several known results of stability for the euclidean spaces to infinite dimensional Banach spaces. The relation between Lemma A and the results in [5, 6, 25] should be clearly understood. Note that in Lemma A it is not supposed that the plane M is the center manifold of the rest point at the origin nor that it is normally hyperbolic. We also note that the semi-complete condition for the solutions of (AS) is a question that should be considered. In particular, we observe that in [3] it is proved that such condition is implied by geometric hypotheses which could be useful in our setting. Finally, we recall that in [23] it is conjectured that for X = (L+H. G) E X m , m 2: 3, X(x,O) = L(x), for every x E m2, with L satisfying
> 0 for every u E m2, (iii) there is p > 0 such that Trace(L'(u)) < 0 provided lIull 2: (ii)Det(L'(u))
p,
Global injectivity and asymptotic stability (iv) h =
357
J JR2Trace(L'(x,y»dxdy f; O.
Then, P oo = (00,0) is a repellor (resp. attractor) for (AS) provided h < 0 (resp. h > 0). We note that such assertion is based on the corresponding result for the two dimensional system proved in [10].
References [1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. F. Anal. 14 (1973), 349-381. [2] K. C. Chang, Infinite dimensional Morse theory and its applications (Birkhauser, Boston, 1993). [3] C. Chicone and J . Sotomayor, On a class of complete polynomial vector fields in the plane, J. of Diff. Eq. 61 (1986), 398-418. [4] A. Cima, A. van den Essen, A. Gassull, E. Hubbers and F. Maiiosas, A polynomial counterexample to the Markus-Yamabe Conjecture, Adv. Math. 131 (1997), 453-457. [5] N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21 (1971/72),193-226. [6] N. Fenichel, Asymptotic stability with rate conditions, Indiana Univ. Math . J. 23 (1973/74), 1109-1137. [7] R. Fesler, A proof of the two dimensional Markus-Yamabe Stability Conjecture, Ann. Polonici Math. 62 (1995),45-75. [8] A. A. Glutsyuk, The complete solution of the Jacobian problem for vector fields on the plane, Comm. Moscow Math. Soc., Russian Math. Surveys 49 (1994), 185-186. [9] C. Gutierrez, A solution to the bidimensional global asymptotic stability conjecture, Ann. Inst. H. Poincare Anal. Non Lineaire 12 (1995), 627671. [10] C. Gutierrez and M. A. Teixeira, Asymptotic stability at infinity of planar vector fields, Bol. Soc. Bras. Mat. (NS.) 26 (1995), 57-66. [11] G. Katriel, Mountain pass theorems and global homeomrphisms theorems, Ann. Inst. H. Poincare, Analyse non lineaire 12 (1994), 189-209. [12] L. Markus, H. Yamabe, Global stability criteria for differential systems, Osaka Math . J. 12 (1960) , 305-317.
358
E. A. B. Silva and M. A. Teixeira
[13] J . Mawhin and M. Willem, Critical point theory and Hamiltonian systems (Applied Mathematical Sciences nO 74. Springer-Verlag, New YorkBerlin-Heidelberg, 1989) . [14] G. H. Meisters, A biography of the Markus-Yamabe Conjecture, Preprint. [15] G. H. Meisters and C. Olech, Solution of the global asymptotic stability Jacobian Conjecture for the polynomial case, in Analyse Mathematique et applications, 373-381, Contributions en l'honeur de Jacques-Louis Lions (Gauthier-Villars, Paris, 1988). [16] G. H. Meisters and C. Olech, Global stability, injectivity and the Jacobian Conjecture, in Froc. of the first world congress on nonlinear analysis, ed. V. Lakshmikanthan (Walter Gruyter & Co .. Tampa. Fl, 1992). [17] C. Olech, On the global stability of an autonomous system on the plane. Cont. to Diff. Eq. 1 (1963), 389-400. [18] T. Parthasarathy, On global univalence theorems, (Lecture Notes in Mathematics nO 977, Springer Verlag, 1983). [19] P. Rabier, On global diffeormophisms of euclidean spaces. Nonlinear Anal.-T.M.A. 16 (1993), 925-947. [20] P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations (C.B.M.S. Regional Confer. Ser. in Math .. nO 65, Am. Math. Soc., Providence. RI. 1986). [21] N. Rouche, P. Habets and M. Laloy, Stability theory by Liapunov's direct method (Applied Mathematical Sciences nO 22, Springer-Verlag, New York-Heidelberg, 1977) . [22] E. A. B. Silva and M. A. Teixeira. A version of Rolle's Theorem and applications, Bol. Soc. Brasil. Mat. (N.S.) 29 (1998), 301-327. [23] E. A. B. Silva and M. A. Teixeira. Global asymptotic stability on euclidean spaces, to appear in Nonlinear Anal.-TMA. [24] M. Struwe, Variational Methods, applications to nonlinear partial differential equations and Hamiltonian systems (Springer-Verlag, BerlinHeidelberg, 1990). [25] S. Wiggins, Normally hyperbolic invariant manifolds in dynamical systems (Applied Mathematical Sciences, nO 105, Springer-Verlag, New York, 1994).
Prescribed energy problem for a singular Hamiltonian system of 2-body type* Kazunaga Tanaka Department of Mathematics, School of Science and Engineering Waseda University 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169, JAPAN
Abstract We consider the existence of a periodic solution of the following prescribedenergy problem: ij + V'V(q) = 0, t1ci12 + V(q) = H .
(HS.l) (HS.2)
Here HER, q(t) : R ~ RN (N ~ 2) and V(q) : RN \ {O} ~ R is a potential with a singularity of 2-body type. More precisely, we assume
(VO) V(q) E C 2 (RN \ {O}, R) . (VI) V(q)
< 0 for
all q E RN \ {O} and V(q) ~ 0 as Iql ~
(V2) There exists an a
> 0 such that V(q)
~ -Iqja near q
00 .
= O.
The order a of the singularity 0 plays an important role for the existence of periodic solutions and it is easily seen that for V(q) = -Iqja , (HS .l)-(HS.2) has a solution if and only if H > 0, if a > 2, H = 0, if a = 2, { H < 0, if a E (0,2). We study (HS .l)-(HS.2) under situations which generalize the above three cases and give existence results. We also study the existence of non-constant closed geodesic on non-compact manifold (R x
sN -1,9)
9(8 ,,,,)
~ ds
where ho is the standard metric on
under a related situation: 2
+ ho
as s ~
±OO,
SN-l
"Dedicated to Professor Paul H. Rabinowitz on occasion of his 60th birthday
359
360
1
Kazunaga Tanaka
Introd uction
In this article we study the existence of periodic solutio'ns of singular Hamiltonian systems as well as the existence of closed geodesics on non-compact Riemannian manifolds in a related situation. For Hamiltonian systems, we consider the second order Hamiltonian system related to the 2-body problem in celestial mechanics, that is, we consider the following equation: (HS.1) it + \7V(q) = 0, where V(q) : RN \ {O} satisfies
-+ R
is a potential with a singularity at 0 E RN and it
(VO) V(q) E C(RN \ {O} , R), (VI) V(q)
< 0 for all q E RN \ {O} and V(q), \7V(q) -+ 0 as Iql-+
00.
(V2) There exists an a > 0 such that 1
V(q) '" - -
IqlO
in C 2 sense, that is, for W(q)
as q'" 0
= V(q) + ~
IqIOW(q), IqIO+1\7W(q), Iqlo+2\72W(q)
-+ 0
as
Iql -+ O.
Clearly the classical Newtonian potential satisfies (V1)-(V3) . For such a potential V(q), we consider so-called prescribed energy problem. That is, for a given HER we try to find a periodic solution of (HS.1) satisfying
~lcjl2 + V(q) = H
(HS.2)
We remark that we don't prescribe the period of solution q(t). For the prescribed period problem we refer to [Bahri-Rabinowitz[6], Ambrosetti and Coti Zelati[2], Coti Zelati[10], Coti Zelati-Serra[ll]' Greco[12]' Tanaka[16]] and references in [Ambrosetti-Coti Zelati[2]], The order a of the singularity 0 plays an important role for the existence of periodic solutions. For example, for a typical case V(q) = - rqf., (HS.1) - (HS.2) has a periodic solution if and only if
H >0
for
a> 2,
(1)
H=O
for
a = 2,
H <0
for
a < 2.
(2) (3)
Prescribed energy problem for a singular Hamiltonian system
361
In fact, if (HS.l)-(HS.2) has a T-periodic solution, we multiply q(t) to (HS.l) and integrate over [0, T] and we get
r 0: r 10 Iql'" dt 10 Iql2 dt. T
T
=
Thus
r (12"lql. HT = 10 T
(0:) r dt = "2 - 1 10
T
2
1)
- Iql'"
1
Iq(t)I'" dt .
Therefore one of (1)-(3) holds. Conversely if one of (1)-(3) holds, we can see that q(t) =R(coswt,sinwt,O,···,O) give a solution of (HS.l)-(HS.2) with a suitable choice of R, w > 0. Periodic solutions of (HS.l)-(HS.2) can be found as critical points of the following functional
r
r
l
I(u) =
l
2"1 10 (H - V(u)) dr 10 lu(r)1 2 dr
(4)
2"1 10r (H - V(u))lu(rW dr
(5)
or
l
J(u) =
Both functionals are defined on the space of I-periodic curves which don't pass through 0: A = {u(r) E Hl(O, 1; RN); u(r) :f:. 0 for all r}. We can easily see that if u(r) is a critical point of I(u) or J(u), after a suitable change of variable q(t) = u(r(t)) gives a periodic solution of (HS.l)-(HS.2). To see the difference of situations (1)-(3), it is helpful to observe the graph of f(t) = I(tu) (or J(tu)) : (0,00) -+ R for u t o. In the setting of (1), f(t) satisfies f(O) = 00, f(oo) = 00 and there exists a unique to > 0 such that f'(to) = 0. In the setting (3), f(t) satisfies f(O) = 0, f(oo) = -00 and there exists a unique to > 0 such that f'(to) = 0 and f(to) > O. Lastly in the setting of (2), we can see that f(t) is a constant function for every u. Since R x SN-l and RN \ {O} are diffeomorphic through a mapping
Kazunaga Tanaka
362
We can reduce our problem to the existence problem for closed geodesics on a non-compact Riemannian manifold R x SN~l with a metric gV defined by grs,z)
= e 2S (H -
(6)
V(eSx»grs,z) '
Here grs ,z) is the standard product metric on R x SN-1 :
grs,z)«~' 17), (~, 17» = 1~12 for (s,x) E R we identify
X
+ 11712
SN-1 and (~ , 17) E T(s ,z)(R x SN-1)
=R
x T z (SN-1). Here
= {17 ERN ; x ' 17 = O} . We remark that g(vs,x ) = gOts,x ) if V(q) = -r::b- and H = O. T z (SN-1)
Iql-
In what follows , we study the existence of periodic solutions under the situations which generalize (1)- (3) and the existecne of non-constant closed geodesics in a situation related to (2).
2
Strong force case
A situation related to (1) is called strong force and it is studied by [AmbrosettiCoti Zelati[l], Benci-Giannoni[8]' Greco[13]' Pisani[14]) . For a large class of V, [Pisani[14]) showed the existence of a periodic solution of (HS.1)-(HS.2) . The following result is a special case of Pisani's result. Theorem 1 (c.f. [Pisani[14)]) Suppose that V(q) satisfies (VO)-(V2) with Then for any H > 0, (HS.l)-(HS.2) has at least one periodic solution.
Q> 2.
Remark 1 Pisani's result is more general. He requires just V(q) E C 1(RN \ {O}, R) and his conditions admit more general behavior at 0 and 00 .
To prove Theorem 1, Pisani uses a minimax method based on the fact 1= 0: essentially he proves the following minimax value is a critical value of J(u) .
7rN- 2(A)
(7) where ~
= {a E C(SN-2,A); dega 1= O},
a(~)(r) .. SN- 2 x([0, l]/{O,I}) '" SN-2 x Sl --* SN-1 . a-('<" r ) -- la(~)(r)1 _ In his argument , a detailed study of t he deformation flow near 0 is important.
Prescribed energy problem for a singular Hamiltonian system
3
363
Weak force case
A situation which generalize (3) is called weak force. This case is studied by Ambrosetti-Coti Zelati[l]' Ambrosetti-Struwe[3]' Coti Zelati-Serra[11]' Tanaka[ Tanaka[18]. In this case we have the existence of a periodic solution rather restricted conditions. The following is a special case of [Tanaka[18]]. Theorem 2 (c.f. [Tanaka[18]]) Suppose N 2: 3 and assume that V(q) satisfies (VO)-(V2) and
4/3 1
Then for any H
< 0,
< a <2 < a <2
if N = 3,
(8)
if N 2: 4.
(9)
(HS.l)-(HS.2) has at lease one periodic solution.
Theorem 2 is unsatisfactory in the following sense, it excludes the case N = 2 and the case a = 1. The difficulty is how to distinguish collision solutions, which is a kind of generalized solution of (HS.l)-(HS.2) that goes through the singularity 0, and non-collision solutions. Here we use a relation between Morse indices and collisions (Proposition 3). To prove our Theorem 2, we apply the following minimax method to I(u) (10)
where
r
=
bE C([ro,rd x SN-2,A); ')'(ro,~) = rolToW, ')'(rl'~) = rllTO(~) for all ~ E SN-2}.
Here lTo E E is fixed and 0 < ro « 1 « rl· Moreover, since our potential is a weak force one, we have difficulties related to the Palais-Smale compactness condition etc. To overcome these difficulties, we introduce a perturbed functional
Ie(u)
=~
11,u,2 11 dT
H - V(u) +
,:,4 dT
and we first apply our minimax method to Ie (u) and try to pass to the limit as f --t O. For a sufficiently small f > 0, by a virtue of our minimax method and the strong force term we can find a critical point U e E A of Ie(u) such that
lW'
(i)
(ii)
Ie(u e) E [m, M], where m, M > 0 are independent of f. (11)
Kazunaga Tanaka
364
Here index I~'(uf) denotes the Morse index of I~'(uf) . It follows from (i) that
where C > 0 is independent of E > o. Thus we can extract a weakly convergent subsequence Ufk(X) -+ uo(x) . However uo(x) may enter the singularity O. To avoid this possibility, we use the property (ii) and the following Proposition 3 If the limit function uo(x) enters the singularity 0, then we
have lim inf index
I~'(uf)
f--tO
(12)
2: (N - 2)i(a),
where i(a) E N is an integer defined by 2
i(a) = {k E N; k < -2- } . -a It satisfies i(a) i(a) i(a)
1 for a E (0,1]' 2
for a E (1,
n
(~, 2).
> 3 for a E
Comparing (11) and (12), we have uo(r) "10 for all r under the assumption of Theorem 2.
4
The case which generalizes (2)
-dr-.
Finally we deal with a perturbation of V(q) = This case is rather different from previous cases and we can expect the existence of periodic solutions only at the level H = o. Thus this situation is rather delicate and the problem (HS .1)-(HS.2) accepts only restricted class of perturbations. A typical feature of the case V (q) = is its scale invariance, that is, the corresponding functionals I(u), J(u) enjoy the following property:
dr-
I(ru) = I(u), J(ru) = J(u)
for all u E A and r > O.
A perturbation problem for scale invariant functionals are studied for semilinear elliptic equations in RN by [Bahri-Li[4], Bahri-Lions[5JJ. Here we use an idea from [Bahri-Li[4]] . In addition to (VO)-(V2) we assume the following condition: (V3) Set W(q) = V(q)
+ dr-,
then W(q) satisfies
IqI2W(q), IqI3\7W(q), IqI4\7 2W(q) -+ 0 as Iql -+
00.
Prescribed energy problem for a singular Hamiltonian system
365
Our existence result is Theorem 4 ([Tanaka[19]]) Assume (VO)-(V2) with Q = 2 and (V3). Then (HS.l)-(HS.2) with H = 0 has at least one periodic solution. The conditions (V2) and (V3) require
V(q) '"
1
-jqj2
as
Iql '" 0 and Iql '" 00.
This condition is necessary for the existence of periodic solutions for (HS.1)(HS.2) with H = 0 in the following sense: if V(q) behaves like
V(q)
as
Iql '" 0,
(13)
V(q)
as
Iql '" 00
(14)
and a -:F b, then (HS.1)-(HS.2) with H = 0 does not have periodic solutions in general. In fact, if V(q) = -
1 r lul 21
=~
2
dr
o
1 r cp(rl~1) dr 1 lui
o
is strictly increasing (or decreasing) function of r for all u ~ 0 and it does not have non-trivial critical points. To apply Bahri-Li's idea to our problem, it is better to use a formulation for closed geodesics on (R x SN-\gV), where gV is defined in (6). Under the conditions (V2) and (V3), gr.,x) satisfies gr.,x) ~ gO
as
lsi
~ 00.
(15)
Here gO is the standard product metric on R x SN-1. We define two minimax values for the following energy functional 1
EV(u) =
~2 1r g~(T)(u,u)dr:
ARxSN-l
o
~ R,
where ARxSN-l is a set of H1-closed curves [0, 1J/{0, 1} :: S1 ~ Rx SN-1. We 1 remark that if V (q) = the corresponding functional EO (u) = f0 IU12 dr is shift invariant, that is,
r!r-,
EO(s(r)
!
+ h, u(r)) =
EO(s(r), u(r))
Kazunaga Tanaka
366
for all U(T) = (S(T), X(T)) E ARxSN-l and hER. The first minimax value Qis defined in spirit of (7). We need some notation to define it. For a (N - 2) dimensional compact manifold M, we set r(M) = bE C(M, A RxS N-2); "'((M) is not contractible in
A R x S N-2}
and M
{M; M is a (N - 2) dimensional compact connect manifold
=
such that
reM) =f. 0}.
We remark that SN-2 E M and M
b(M) =
=f. 0. For M
inf
E
M, we define
maxE v b(~))
')'Er(M) ~EM
and
b = inf b(M).
-
MEM
We can observe that 0 < Q~ 211'2. The second minimax value b is defined in spirit of (10). We define
r = Here 0'0 E
~
bE C(R x SN-2,A R x S N-2); "'((r,~)(t) = (r,O'o(~)(t)) for sufficiently large Ir I}·
satisfies
I0'o (~) (T) I =
1 for all
~,
T and
We set b=in[ ')'Er
sup
EVb(r,~)) .
(r , ~)ERxSN-2
We can show b :::: 211'2. One of the following 3 cases occurs.
(i) (ii) (iii)
0 < Q<
211'2,
< b, Q = b = 211'2. 211'2
As in [Bahri-Li[4]), we can see Q (b, 211'2 respectively) gives a critical value of E in case (i) ( (ii), (iii) respectively) . We remark that 211'2 is the minimal critical value of unperturbed functional EO(u) and the corresponding solution is the great circle. Key of the proof of Theorem 4 is the following Lemma, which describes the break down of the Palais-Smale condition.
Prescribed energy problem for a singular Hamiltonian system
367
Lemma 5 Under the condition (15), the Palais-Smale condition for E V (u) breaks down only at the levels c E {27r 2 k 2 ; kEN}. Moreover non-convergent sequence (Uj)~l = (Sj, Xj)~l C ARxSN-' satisfying EVI(Uj) ~ 0 and E V (Uj) c = 27r 2 k 2 has a subsequence - still denoted by Uj - such that (i) Sj(O)
~ 00
or Sj(O)
~ -00,
(ii) uj(r) == (sj(r) - Sj(O),xj(r)) ~ (O'Yk(r)) in ARxSN-l as j ~
00,
where
(iii) EV(uj) ~ 27r 2 k 2 . (iv) liminfj_+
Theorem 6 ([Tanaka[19]]) Let 9 be a Riemannian metric on R x SN-l and suppose that 9 satisfies 9 '" gO as S '" ±oo. More precisely, (gO) 9 is a C 2 -Riemannian metric on R x SN-l. (g1) 9 '" gO as s '" ±oo in the following sensei let (e,···,~N-l) be a local coordinate of SN-l in an open set U C SN-l and set ~o = s. We write N-l 0 1 , ... ,~ N - 1 · . g= "L gij(~,~ )d~·Q9de· i,j=O We also write gO = Eg?j(~O,e,···,~N-l)deQ9d~j, where gO is the standard product Riemannian metric on R x SN-l. We remark that g?j(~O, e, ... , ~N-l) is independent of ~o = s. We assume gij(S,e, ... ,~N-l)
~
8:;j(s,e,···,~N-l) ~
g?j(e,···,~N-l)
0
inC1(U,R)
in C 2 (U,R), (16) as
lsi ~ 00.
(17)
Then (R x SN-l) has at least one non-constant closed geodesic .
Remark 2 For the existence of closed geodesics on non-compact Riemannian manifolds, we also refer to [Thorbergsson[20j, Bangert[7j, Benci-Giannoni[9jJ.
368
Kazunaga Tanaka
Acknowledgments The Author would like to thank the organizers of the meeting for their warm hospitality.
References [1] A. Ambrosetti and V. Coti Zelati, Closed orbits of fixed energy for singular Hamiltonian systems, Arch. Rat. Mech . Anal. 112 (1990), 339-362. [2] A. Ambrosetti and V. Coti Zelati, Periodic solutions of singular Lagrangian systems, Birkhiiuser. Boston, Basel, Berlin, 1993. [3] A. Ambrosetti and M. Struwe, Periodic motions for conservative systems with singular potentials, NoDEA Nonlinear Differential Equations Appl. 1 (1994),179-202. [4] A. Bahri and Y. Y. Li, On a min-max procedure for the existence of a positive solution for certain scalar field equations in R N , Revista Mat. Iberoamericana 6 (1990), 1-15. [5] A. Bahri and P. L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. lnst. Henri Poincare, Analyse non lineaire 14 (1997), 365-413. [6] A. Bahri and P.H. Rabinowitz, A minimax method for a class of Hamiltonian systems with singular potentials, J. Punct. Anal. 82 (1989),412-428. [7] V. Bangert, Closed geodesics on complete surfaces, Math. Ann. 251 (1980),83-96. [8] V. Benci and F. Giannoni, Periodic solutions of prescribed energy for a class of Hamiltonian systems with singular potentials, J. Diif. Eq. 82 (1989), 60-70. [9] V. Benci and F. Giannoni, On the existence of closed geodesics on noncompact Riemannian manifolds, Duke Math. J. 68 (1992), 195-215. [10] V. Coti Zelati, Periodic solutions for a class of planar, singular dynamical systems, J. Math. Pure Appl. 68 (1989), 109-119. [11] V. Coti Zelati and E. Serra, Collisions and non-collisions solutions for a class of Keplerian-like dynamical systems, Ann. Mat. Pura Appl. (4) 166 (1994), 343-362.
Prescribed energy problem for a singular Hamiltonian system
369
[12] C. Greco, Periodic solutions of a class of singular Hamiltonian systems, Nonlinear Analysis; T .M.A . 12 (1988), 259-269. [13] C. Greco, Remarks on periodic solutions, wit~ prescribed energy, for singular Hamiltonian systems, Comment. Math . Univ. Carolin . 28 (1987) , 661-672. [14] L. Pisani, Periodic solutions with prescribed energy for singular conservative systems involving strong force, Nonlinear Anal: T .M.A. 21 (1993), 167-179. [15] E . Serra and S. Terracini, Noncollision solutions to some singular minimization problems with Keplerian-like potentials. Nonlinear Anal.:T .M.A . 22 (1994), 45-62. [16] K. Tanaka, Non-collision solutions for a second order singular Hamiltonian system with weak force, Ann. Inst . Henri Poincare, Analyse non lineaire, 10 (1993), 215-238. [17] K. Tanaka, A prescribed energy problem for a singular Hamiltonian system with a weak force, J . Punct. Anal. 113 (1993), 351-390. [18] K. Tanaka, A prescribed-energy problem for a conservative singular Hamiltonian system, Arch. Rational Mech . Anal. 128 (1994), 127-164. [19] K. Tanaka, Periodic solutions for singular Hamiltonian systems and closed geodesics on non-compact Riemannian manifolds, to appear in Ann. Inst .
Henri Poincare, Analyse non lineaire [20] G. Thorbergsson, Closed geodesics on non-compact Riemannian manifold, Math . Z. 159 (1978), 249-258.
Sign-Changing Solutions for a Class of Nonlinear Elliptic Problems Zhi-Qiang Wang • Department of Mathematics, Utah State University, Logan, UT 84322, USA E-mail: wang(lmath. usu. edu.
(Dedicated to Professor P. Rabinowitz on the occasion of his 60th birthday)
1. Introduction For semilinear elliptic boundary value problems, great progress has been made in the last three decades. The research in this field has exploded, following the most celebrated paper [5] of Ambrosetti and Rabinowitz in which they prove the Mountain Pass theorem, which has motivated and inspired the revolutionary development of minimax theory and Morse theory (e.g., [3] [11] [15] [25] [39] [43]). We are in no position to survey the field, rather we will concentrate on a subarea which has not been given much attention until quite recently, namely, the existence and qualitative properties of sign-changing solutions for elliptic boundary value problems. Looking back in the literature we see much more attention has been given for the study of positive solutions (e.g., [2] [19] [35] [21]). First, the structure of positive solutions for an elliptic problem is relatively simpler than that of sign-changing solutions. Many problems desire finitely many positive solutions (signed solutions) (e.g., [29]) if not unique, while this is in general not the case for sign-changing solutions. Consider -~u + f(u) = 0, x E { u = 0, x E an,
n,
(1.1)
where n c RN (N ~ 1) is a bounded domain with smooth boundary assume fECI (RN , R) and make some further assumptions.
(fd If(t)1 :::; C(1 + lti',,-l) , 2 < Q < 2'
:= ~::'2'
(h) There exist f.L > 2 and M > 0 such that with F(t)
o < f.LF(t) (h) f(O)
:::; tf(t) ,
= 1'(0) = o.
• Supported in part by a NSF grant .
370
It I ~
M.
= J~ f(s)ds
an.
We
Sign-changing solutions for nonlinear elliptic problems
371
Theorem 1.1 ([5]) Under (ft), (h) and (h), (1.1) has a positive solution and a negative solution. If in addition, (/4) f( -t) = - f(t) .
holds, then (1.1) has infinitely many solutions.
Note here that though the number of signed solutions may vary depending upon the domain and nonlinearity (e.g., [20] [9] [26]), generally speaking, one would expect a finite number of signed solutions of (1.1). Thus among the solutions of (1.1) we would expect infinitely many sign-changing solutions. This is the case for ordinary differential equations even without (14) where one may count the number of solutions according to the number of zeroes of solutions (e.g., [38] [28]), and this is also verified recently in [6] for PDEs when (14) is assumed. Hence a natural open question is whether (1.1) has infinitely many sign-changing solutions under only (ft), (h) and (h)· On the other hand, the nodal structure of solutions becomes an interesting subject to study, even for linear problems. A classical result of Courant of [18] states that the number of nodal sets of an eigenfunction of the Laplacian in 0 is bounded above by the Morse index of the eigenfunction. In general, one can not expect a simple relation as above between the Morse index and the number of nodal sets for solutions of nonlinear elliptic problems. Also of interest with sign-changing solutions are the number of nodal regions, the shape of nodal regions relative to the shape of the whole domain O. In recent years, progress has been made for these interesting respects related to sign-changing solutions. In this note, we shall survey some results along the line of study for equation (1.1) and pose some questions. The basic machinery of treating sign-changing solutions is the cone structures in HJ(O) and CJ(O) . Taking advantages of these structures, Amann, Hofer, and Dancer have studied multiplicity of positive solutions. In [2], [19] [21], degree theory and index theory in positive cones were used and in [26] critical points theory was used in partially ordered Hilbert spaces. In [7] [8], critical points theory in partially ordered Hilbert spaces has been further developed to treat sign-changing solutions, and most recent results of existence and multiplicity of sign-changing solutions presented here are based on this framework ([7] [8] [30] [31]). The paper is organized as follow. In section 2, using (1.1) as a model equation we shall review recent results on sign-changing solutions, giving ideas and sketching the basic methods in treating sign-changing problems. In section 3, we present a related problem in which the nonlinearity is not order preserving even under linear perturbations. With a nonlinear perturbation we construct a flow that possesses a certain invariance property and a deformation property.
Zhi-Qiang Wang
372
Some interesting questions are raised throughout the article.
2. On a superlinear elliptic problem Consider equation (1.1) under the assumptions (h), (fz) and (h). Improving the result in [5], we proved in [42] Theorem 2.2 {[42}} Under (h), (fz) and (h), (1.1) has at least three nontrivial solutions, including a positive and a negative one. Remark 2.3 We gave in {42} two different arguments for the existence of the third solution, one based on Morse theory and the other a geometrical linking argument. Under much stronger conditions than ours, a result of similar spirit was given in {40} by Struwe using different method, but with a gap (which was fixed in a private communication). Under stronger conditions than ours, Ambrosetti, Azorero and Peral ([4j) gave yet another different proof of the third solution.
In general, the third solution should be a sign-changing solution since (1.1) may have only one positive and one negative solution. This was confirmed in
[7]. (15) There is a > 0 such that f'(t) > -a for all t. Theorem 2.4 ([7j) Under (ld, (fz), (h) and (15) , (1.1) has a sign-changing solution w.
The idea is based on a modified version of the geometrical linking which was used in [42] . We give a sketch here since it is quite simple. Let us define the functional whose critical points are solutions of (1.1)
and the negative gradient flow cpt of 4.> on E = HJ(D), that is, ftcpt = - '\74.> 0 cpt { cpo = id .
(2 .1)
Let us iqtroduce some notations. Let PE C E be the positive cone in E. As usual, t~is turns E into a partially ordered space where u
~
v
:{=:}
u - v E PE, u> v
:{=:}
u
~
v and u
i- v .
Sign-changing solutions for nonlinear elliptic problems
373
A map f : E -+ E is called order preserving if
u ~v
=> f(u)
~
f(v)
for all
u, vEE.
Let X = CJ (0) c E, which is densely embedded into E. We set P := X n PE and then P has nonempty interior P :f:. 0. Thus X is a partially ordered Banach space and we define u » v :¢::=} u - v E P . The elements of P are called positive, those of -P negative. A map f : X -+ X is said to be strongly order preserving if
u
> v => f(u)>> f(v) for all u,v EX.
First we state the following lemma from [7] which is the basis of the argument. Note that the gradient of if? is of the form 'V if? = I d - K E where K E : E -+ E is a compact operator. Moreover, KE(X) C X and the restriction K := KEIX : X -+ X is of class Cl and strongly order preserving. Lemma 2.5 Suppose if?/(UO) = 0, so that Uo EX. Then for every v E P - {O} and every t > 0 we have cpt (uo ± v) E Uo ± P. Consequently Uo ± P and Uo ± P are positively invariant. Sketch of proof of Theorem 2.4. With an approximation argument given in [8], without loss of generality we may assume if? is C 2 and satisfies the (PS) condition. Let Vo E P be the unique normalized positive eigenvector of if?1I(0) = Id - DKE(O) which spans the one-dimensional eigenspace belonging to the largest eigenvalue of DKE(O). By (h), there exist R > 0 and two orthonormal vectors voo , Woo E X such that Voo E P and if?(u) < 0, if?/(U)U < 0 for every u E span{voo , woo} with lIull ~ R. Then we set
T := {tvoo : -R ~ t ~ R} U {R(v oo cosO + Woo sinO) : 0 ~ 0 ~
7r}
and C := conv(T), the convex hull of T. Clearly T is homeomorphic to S1 and C to B2 . An easy degree argument shows that (C, T) and S link. By this we mean that for every continuous deformation h t : C -+ X with ho(u) = u for u E C, ht(T) n S = 0 for t E [0,1]' it follows that ht(C) n S :f:. 0 for t E [0,1] . From this we deduce that the w-limit set of C has nonempty intersection with S, so there exists v E w( C) n S. From the construction of S it follows that cpt (v) -+ 0 as t -+ 00. In addition, v E w(C) implies limH-oo if? (cpt (v)) ~ maxif?(C) . As a consequence of the Palais-Smale condition there exists a critical point w in the a-limit set of v . This has the required properties. In fact, it cannot be
Zhi-(.Jiang wang
374
comparable to 0 because U1 E a( v) n (F U (- F)) would imply v E F U (- F) by Lemma 2.5. This is not possible since v ESC X \ (P U (-P)) . o. Theorem 2.4 does not give any information about the energy, the Morse index, the nodal property of the solution. Using a beautiful minimization argument, Castro, Cossio and Neuberger ([12]) gave the following result under some additional conditions.
liP- for t '" o. (17) limltl-+oo liP- = (f6) f'(t) >
00.
Theorem 2.6 ([12]) Under (h)'(h), (h), (f6) and (17), {1.1} has a signchanging solution w such that w changes sign exactly once in n. If nondegenerate, w has Morse index 2. Furthermore, w is the least energy sign-changing solution. The method in [12] is to use the Nehari manifold:
N where ')'(u) = (cp'(u),u) =
= {u E E \ {O} h(u) = O}
In lV'ul
N1 = {u E N
2
-
uf(u). Then they define a subset of N
I u+ '" O,u_
'" O,')'(u+) = O} .
They characterized the energy level of w, by c = inf <JI(u) , Nl
and showed c is always achieved at some w which a critical point of <JI in X .
Remark 2.7 Methods of similar spirits to the above has been used in [13} [41} and also [25} to treat critical exponent problems. Quite recently, using yet another different method, namely, homological linking method, we have greatly improved the above result.
Theorem 2.8 ([8]) Under (h), (h), (h) and (fs), {1.1} has a sign-changing solution w which is neither a local minimum nor of mountain pass type. If in addition (/6) holds and sign-changing solutions are isolated, then w has precisely two nodal sets, it has Morse index 2 and its critical groups are like those of a nondegenerate critical point of Morse index 2: Cq(cp,w) = 6q2 G.
Sign-changing solutions for nonlinear elliptic problems
Recall that the critical groups of an isolated critical point Uo with are defined by
375
~(uo)
= (
Here Hk denotes singular homology with coefficients in a field G. We say w iE of mountain type in the sense of [27] if for any small open neighborhood N oj Uo the set ~ c n N \ {uo} , c = ~(uo), is not path-connected. The proof uses a similar linking as in the proof of Theorem 2.8, but in termE of homological language. Sketch of a proof of Theorem 2.8. We describe the construction of thE critical value here only. Since (h) holds, there exists Voo E P and wool.voo with IIv oo II , Ilwooll = 1 and such that ~(u) < 0 for any u E span {voo,woo } with Ilull ~ R, where R > 0 is large. Now we set B := {svoo + twoo : lsi:::; R,
0:::; t :::; R}
and 8B := {svoo
+ twoo
E B : lsi = R or t E {O, RD ·
By construction we have 8B C ~o U D, where D = P U (-P). Let (3 := max ~(B) so that (B,8B) Y (~,8 U D, ~o U D) . Let ~,8 E H2(~,8 U D , ~o U D) be the image of 1 E G ~ H 2 (B,8B) under the homomorphism
induced by the inclusion. For 'Y :::; (3 let
be also induced by the inclusion. Now we define
and c:= infr. An argument in [8] gives ~,8 ;j; O. We have 0 rt r because jo = o. Clearly (3 E r, hence c E [0, (3] . Morover, by (h) the sign changing solutions cannot accumulate at O. Assuming that the sign changing solutions are isolated WE see that there can only be finitely many sign changing solutions with values in [0, (3]. This implies that ~o U D is a strong deformation retract of ~'Y U D fOJ 'Y > 0 small enough, and consequently c > o. From here one may argue as in [8] to get the conclusions. 0
376
Zhi-Qiang Wang
Remark 2.9 The method for proving Theorem 2.8 above also is set up in an abstract framework and can be used in more general problems and with other type of nonlinearities as well as nonlinear elliptic systems {see [8} for details) . Remark 2.10 Using Conley index theory, Dancer and Du have given some results in [22} [23} in similar spirit. Using invariant sets under descending flow , Liu ([36]) and Li and Zhang ([32}) also have produced similar results. But none of these results give the information about critical groups and Morse indies as in our Theorem 2.8. Remark 2.11 An interesting question is to find the location of the nodal sets of wand the profile of w relative to the domain n. Remark 2.12 Another related result is for the symmetric case. In [6}, under the additional condition (f4) infinitely many sign-changing solutions were proved. A natural question here is whether there exist infinitely many signchanging solutions whose numbers of nodal sets tend to 00 . With all these efforts, it is still an open question that whether there are infinitely many solutions (or infinitely many sign-changing solutions) without assuming any symmetry.
3. Some related results There are some variants of the above results. Remark 3.13 If 1'(0) > >'2, and there are two pairs of strict sub-super solutions, then one can get seven nontrivial solutions including two pairs positive and negative solution, and three sign-changing solutions {e .g., [14} [22} [32]). Remark 3.14 There have been some work in numerically obtaining solutions of saddle points type {e .g, [17} [16]) . Following [17} which numerically calculated mountain pass solutions (signed solutions) there have been some efforts in numerically capturing sign-changing solutions. Recently, many numerical results have appeared in supporting the conjecture that under only (II) , (h) and (h), (1.1) has infinitely many solutions. Based on the linking construction in [42}, a numerical algorithm has been given by Ding, Costa and Chen in [24} to construct many sign-changing solutions. Different numerical algorithms are also given in [34} [33} to capture sign-changing solutions.
Sign-changing solutions for nonlinear elliptic problems
377
Next, we consider a related problem to (1.1) -.6.u = AU + h(x)f(u), x E 0, { u = 0, x E a~,
(3.1)
where h(x) E G"'(O) is a sign-changing function. We use the following assumption from [1], which is stronger than (fd and (h).
(fa) For some a, 2 < a < 2*
(hd Suppose that h has a "thick" zero-set {xlh(x)
> O} n {xlh(x) < O}
=
0.
Under (/3) and (fa) and (hI)' among other things, Alama and Del Pino [1] proved the existence of three nontrivial solutions of (3.1) (including a positive and a negative solution) provided A < Al (0). A related problem was also studied in [10] . Here we denote the eigenvalues and eigenfunctions of -.6. on 0 by Ai E a( -.6.) and cpi(X) . This was done using similar arguments to those in [42] . The following result generalizes the above mentioned result from [1] and gives nodal properties of solutions. Theorem 3.15 ([30]) Assume (/3), (f5), (fa) and (hI) hold. If A < Al (0), then {3.1} has at least three nontrivial solutions, one is positive, one is negative and the third is sign-changing.
Under lightly stronger conditions, we have the following result in [30]. As in [37] we define
A* = inf{
II\7vll~ IIIvl12 =
1,
k
h(x)lvIPdx = O}.
Theorem 3.16 ([30]) Assume (/3), (f5), (fa) and (hd hold. In addition we assume
(19) For some q> 2 it holds limu-+o IJJ~t (h2)
= b > O.
In h(X)(CPI(X))qdx < o.
Then there exists X> 0, X::; A*, such that if A2(0) < A < X and A f/. a( -.6.), {3.1} has at least seven nontrivial solutions. More precisely,
378
Zhi-Qiang Wang
(a) (3.1) has at least two positive solutions ui and ut where 4>(ut) < 0, ut is a local minimizer of 4>(u) . (b) (3.1) has at least two negative solutions u 3 and ui where 4>(ui) < 0, ui is a local minimizer of 4>(u). (c) (3.1) has at least three sign-changing solutions U5,U6 and U7 where U6 is a mountain pass point of4>(u), 4>(U6) < 0, ui < U6 < ut, ui < U7 < ut , and U7 is outside [ui, ut] .
Remark 3.17 If only>,! (n) < A < X and A (j. 0'( -~) are assumed, then (3.1) has at least five nontrivial solutions. More precisely, (a) and (b) still hold and there is a sign-changing solution U5 . This gives the nodal information of a result in [1].
Looking back at the proofs in the last section, we see our whole approach is based on the validity of Lemma 2.5. The negative gradient flow leaves the positive and negative cones invariant and ci(±P \ {O}) C ±P for t > O. However, in problem (3.1), the difficulty is that with the presence of hex) the nonlinear term is not order preserving any more, even under linear perturbations. In [30] we constructed a flow which may not be a pseudo gradient flow in general, but which still has some invariant property with respect to the positive and negative cones. This is done with an abstract framework and should be useful in other problems. We give a brief description of this construction. Let us make the following assumption.
(4)t) 4> E C 2 (E, R) and the gradient of 4> is of the form \74>(u) = Ao(u) -KE(U) where KE : E -+ E is compact, KE(X) C X and the restriction KE on X, K : X -+ X is of class C 1 and is strongly order preserving, i.e .. u > v => K(u) » K(v) for all u, v E X, where u »v ¢:> u - v E P; Ao : E -+ E is a locally Lipschitz operator such that Ail 1 : E -+ E exists, Ail 1 (X) c X and the restriction Ail 1 1x : X -+ X is of class Cl and strongly order preserving. Furthermore, there is M > 0 such that Ao(u) is linear for all u E X and Ilullx ::; M . Finally, we assume that there exists ao > 0 such that
and
Under the above assumption (4)1) ' if 4> also satisfies the (PS) condition, we shall construct a flow 1](t, u) along which 4> decreases and satisfies a deformation property, and under which ±P and ±P are both positively invariant, and finally
Sign-changing solutions for nonlinear elliptic problems
379
1l( t, u) leaves any order interval of the form [y., u) invariant provided Y., u is a pair of sub and super solutions with 1Iy.llx ~ M and Ilulix ~ M . Here we define Y. (u, resp.) a sub (super, resp.) solution if K(y.) > Ao(y.) (K(u) < Ao(u), resp.). Consider dtT~tt) = -Ail l (\7(O'(t,u))) {
0'(0, u)
= u.
Lemma 3.18
O'(t, ±P \ {O}) C ±P o 0 O'(t, ± P \ {O}) C ± P { O'(t, [y., u)) C [y., u) provided
1Iy.llx
~
M,
lIulix
~
'tit> 0,
"It> 0, "It> 0
M, and K(y.) > Ao(y.) and K(u) < Ao(u).
Proof. Let v E P \ {O} and consider v - Ail l (\7(V)). It suffices to show v - Aill(\7(V)) »0. This is equivalent to Ao(v) » \7(v) = Ao(v) - K(v) which follows from K(v) »0. Here we use the fact Ail l is strongly order preserving. Now, for any v E P such that y. + v ~ u,
follows from
which holds if
Ao(v) since K(y' + v) linear, we get
»
+ Ao(y.) > Ao(y. + v)
K(y') > Ao(y.). By the assumption, for
lIulix
~
M, Ao(u) is
o Remark 3.19 Note that -Ail l (\7(u)) is not a pseudo gradient vector field
for -\7(u), in general. We only get ±P are positively invariant. We do not know whether Uo ± P is positively invariant for any critical point Uo, like in Lemma 2.5. However, we still get the deformation property for using this flow. The following lemma is modeled on a deformation lemma in {43}.
Zhi-Qiang Wang
380
Lemma 3.20 Let SeE,
C
E R, £0
> 0, f> >
°such that
2£0 Vu E 4>-I([C - 2£0; C + 2£0]) n S2,s ~ 11V'4>(u)1I ~ ~. uao
Then 37] E C([O, 1] x E, E) such that (i) 7](t,u) = u, ift = or ifu r{. 4>-I([C - 2£O,c + 2£0]) n S20 . (ii) 7](1, 4>C+E n S) c 4>C-E. (iii) 7](t,·) is an homeomorphism of E, Vt E [0,1]. (iv) 117](t, u) - ull :::; f>, Vu E E, \:It E [0,1] . (v) 4>(7](t,u)) is nonincreasing, Vt E (0,1] . (vi) 4>(7](t, u)) < c, Vu E 4>c n S,s, Vt E (0,1]. (vii) 7] has the property of Lemma 2.1.
°
7](t, u) is built out of aCt, u), and for a proof of this lemma, see [30]. Now let us briefly describe how to apply this procedure to (3.1). Under (/5) and without hex), the usual method is to add a linear term on two side of the equation. Since h changes sign, this method fails. Our idea here is to add a nonlinear term on two sides of the equation. Let us define a function m(u) such that for lui:::; Mo for lui ~ Mo + 1 and m(u) is C 1 and monotonically increasing. Here, mo, Mo and p are constants which can be fixed according to feu) and hex) and the needs in the applications. Now consider
{
-/:::,.u + m(u) u
= AU + h(x)f(u)
+ m(u)
in n on an.
:= f(x, u)
=0
We choose Q < p < 2*, thus if mo and Mo are large, f(x, u) is strictly increasing. Let
4>(u)
=~
In
2
lV'ul dx
+
In
M(u)dx
-In
F(x, u)dx,
u E E,
where M(u) = Jou(x) m(s)ds . Then V'4> has a form
V'4>(u) where K(u)
= (_/:::,.)-1 f(x , u)
= Ao(u) -
K(u)
is strongly order preserving, and
(Ao(u) , v)
=
In
(V'uV'v
+ m(u)v)dx
Sign-changing solutions for nonlinear elliptic problems
381
i.e., Ao(u) = u + (-L:,)-lm(u) . Note that Ao(u) is linear for Ilullx ~ Mo. It is easy to see Ao E C l and Ao has a Cl inverse map Aill. By the above formula,
If u
= v, we get (u, Aol(u))
2: IIAill(u)112 .
The other requirements in (
References 1. S. Alama and M. Del Pino, Solutions of elliptic equations with indefinite nonlinearities via Morse theory and linking, Anal. Nonlineaire 13(1996) 95-115. 2. H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Review 18(1976) 620- 709. 3. A. Ambrosetti, Critical points and nonlinear variational problems , Mim. Soc. Math . France (NS.), No. 49 (1992). 4. A. Ambrosetti, J . Azorero and 1. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal., 137 (1996) , 219-242. 5. A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications, Jour. Funct. Anal. 14(1973) 349- 381. 6. T. Bartsch, Critical point theory in partially ordered Hilbert spaces, preprint. 7. T . Bartsch and Z.-Q. Wang, On the existence of sign changing solutions for semilinear Dirichlet problems, Topo. Meth. Nonl. Anal. 7(1996) 115-131. 8. T. Bartsch, K.C. Chang, Z.-Q . Wang, On the Morse indices of signchanging solutions for nonlinear elliptic problems, Math. Zeits. , to appear. 9. V. Benci and G. Cerami, The effect ofthe domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. Rat. Mech . Anal., 114 (1991), 79-93. 10. H. Berestycki, 1. Capuzzo-Dolcetta and L. Nirenberg, Super linear indefinite elliptic problems and nonlinear Liouville theorems, Top . Meth . Nonl. Anal. 4(1994) 59-78.
382
Zhi-Qiang Wang
11. H. Brezis and L. Nirenberg, HI versus Cl minimizers, C.R.Acad. Sci. Paris, 317 serie I (1993) 465-472. 12. A. Castro, J . Cossio, J .M. Neuberger, A sign changing solution for a superlinear Dirichlet problem, Rocky Mount. J. Math., 27 (1997), 10411053. 13. G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal., 69(1986), 289-306. 14. K.C. Chang, Morse theory in nonlinear analysis, Proc. Symp. ICTP (1997), to appear. 15. K.C. Chang, Infinite dimensional Morse theory and multiple solution problems, Birkhauser Boston (1993). 16. G. Chen, W. Ni and J. Zhou, Algorithms and visualization for solutions of nonlinear elliptic equations, Part I: Dirichlet problems, preprint. 17. Y.S. Choi and P.J. McKenna, A mountain pass method for the numerical solution of semilinear elliptic problems, Nonl. Anal., 20(1993), 429-437. 18. R. Courant and D. Hilbert, Methoden der Mathematischen Physik I. Berlin: Springer 1924. 19. E.N. Dancer, On the indices of fixed points of mappings in cones and applications, J. Math. Anal. Appl., 91 (1983), 131-151. 20. E.N. Dancer, The effect of domain shape on the number of positive solutions of certain nonlinear equations, J. Diff. Equs. 74(1988), 120-156. 21. E.N. Dancer, Positivity of maps and applications, Progr. Nonl. Diff. Equs. Appl., 15(1995) 303-340, Birkhauser, Boston. 22. E.N. Dancer and Y. Du, A note on multiple solutions of some semilinear elliptic problem, J. Math. Anal. Appl. 211(1997) 626-640. 23. E.N. Dancer and Y. Du, The generalized Conley index and multiple solutions of semilinear elliptic problems, Abs. Appl. Anal. 1:1(1996) 103-135. 24. Z. Ding, D. Costa and G. Chen, A high liking method for sign-changing solution for semilinear elliptic equations, Nonl. Analysis, to appear. 25. N. Ghoussoub, Duality and perturbation methods in critical point theory, Cambridge University Press, Cambridge, 1993. 26. H. Hofer, Variational and topological methods in partially ordered Hilbert spaces, Math. Ann. 261 (1982), 493 - 514. 27. H. Hofer, A note on the topological degree at a critical point of mountain pass type, Proc. AMS 90(1984) 309-315. 28. H. Jacobowitz, Periodic solution of x" + g(t, x) = 0 via the PoincareBirkhoff theorem, J. Diff. Equs., XX(1976) 37-52.
Sign-changing solutions for nonlinear elliptic problems
383
29. M.K. Kwong, Uniqueness of positive solutions of ~u+u+uP = 0 in R n , Arch. Rat. Meeh . Anal., 105 (1989), 243-266. 30. S. Li and Z.-Q . Wang, Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems, Jour. d'Analyse Math ., to appear. 31. S. Li and Z.-Q. Wang, Dirichlet Problem of Elliptic Equations with Strong Resonances, preprint. 32. S. Li and Z.T. Zhang, Sign-changing solutions and multiple solutions theorems for semilinear elliptic boundary value problems, to appear. 33. Y. Li and J. Zhou, A minimax method for finding critical points with general Morse index and its applications to semilinear PDE, preprint. 34. J. M. Neuberger, A numerical method for finding sign-changing solutions of superlinear Dirichlet problems, preprint. 35. P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Review, 24(1982), 441-467. 36. Z. L. Liu, Multiple Solutions of Differential Equations, Ph. D. Thesis, Shandong University, Jinan, 1992. 37. T. Ouyang, On the positive solutions fo semilinear equations ~u +.AU + hu P = 0 on compact manifolds, Part II, Indiana Univ. Math. J., 40 (1992) 1083-1140. 38. P. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Roeky Mountain Math. J., 3 (1973), 161-202. 39. P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations. CBMS Reg. Conf. Ser. in Math. 65, AMS, Providence R.I. 1986. 40. M. Struwe, Three nontrivial solutions of anticoercive boundary value problems for the pseudo-Laplace operator, J. Reine Angew. Math. , 325 (1981), 68-74. 41. G. Tarantello, Nodal solutions of semilinear elliptic equations with critical exponent, DijJ. Int. Equs., 5(1992), 25-42. 42. Z.-Q. Wang, On a superlinear elliptic equation, Analyse NonLineaire 8(1991), 43 - 57. 43. M. Willem, Minimax Theorem, Birkhiiuser (1996).
A class of resonant or indefinite elliptic problems * Shaoping Wu Department of Mathematics, Zhejiang University Hangzhou, 310027 China Abstract We study the elliptic problem -~u
{ ulao
= AU = 0,
a(x)lulq-1u + f(x, u),
in
0
(D)
where 0 is a bounded domain in RN(N ~ 3) with smooth boundary, the function a(x) E Loo,O < q < 1, and the function 9 is sublinear at infinity, while A is a parameter in R 1. Some existence and multiplicity of the nontrivial solution for Eq.(i E N),(D), are obtained as A ~ Ai(i EN), where Ai is the i-th eigenvalue of -~ in the space E = HJ(O). Also a bifurcation phenomenon is found and a homogeneous case is studied. Finally a sign changing solution is found .
1. Introduction Let 0 be a bounded domain in RN (N ~ 3) with smooth boundary. We study the elliptic problem -~u = AU - a(x)lulq-1u + f(x, u), { ulao = 0,
in
0
(D)
where a(x) E Loo, 0 < q < I, and the function f is sublinear at infinity, while A is a parameter. It comes from some problem in population dynamics [1) . As a(x) > 0, Eq. (D) is discussed in [2] with some monotone condition. The existence and uniqueness of a positive solution is proved by a monotone argument Key words and phrases. sublinear term, semilinear elliptic equation, resonant case • supported in part by National Natural Science Foundation of China and Zhejiang Natural Science Foundation
384
A class of resonant or indefinite elliptic problems
385
combined with a minimizing procedure. Moreover, the existence of infinitely many nontrivial solutions can be obtained by the oddness of the nonlinear term according to the index theory. Later as a(x) = A > small and combined with a superlinear term g(x, u) = uP, a second positive solution is obtained by means of Mountain Pass Lemma in [3] . The effect of the local behaviour of sublintar term at zero on Eq.(D) is studied in [4,5] . It is interesting to find more about the effect of the sublinear term at zero on the elliptic problems. The problem is called indefinite as A > Al and called resonant as A = Ai(somei EN). There is a lot of work concerning this kind of problem by different methods, especially as the nonlinear function is C 1 . For example, see [6-9] and the refernces there. However, the nonlinear function a(x)luI Q- 1 u + f(x, u) is not only sublinear at infinity but also at the origin. The nonlinear term on the right hand side is only Harder continuous at origin and the linearized method can not be applied directly. So is the implicit function theorem. It also brings some difficulty in getting priori estimate for the solution. By means of the spectral theorem and the Mountain Pass Lemma and its variants [10] we present some existence and multiplicity of the solution for Eq. (D). The following assumptions are made:
°
(ad
The function a E LOO(O) .
(fd The function f E C(O b1 , b2 > such that
°
X
Rl,RI) and there are constants p E (0,1) and
If(x, s)1 ::; bdsl P + b2 ,
°
(h) f(x,O) = and f(x,s)s > Our main results in [11] are
°
for
for s -::F
x E 0,
°
s E RI
and (x,s) EO
X
RI.
Theorem 1 Suppose the conditions (ad with a(x) > 0, and (II), (h) hold with 2p < q + 1. Then, for any given A > AI, the Eq. (D) has a nontrivial solution in space HJ (0). In addition the function f is odd in s, the Eq. (D) has m pairs of nontrivial solutions, where m = dimE I + ... + dim E i - 1 , as A = Ai ; m = dim El + ... + dim E i , as A E (Ai, AHd, where i E Nand Ei is the i-th eigenspace of the operator -~ in HJ. Corollary 2 Under the assumptions of Theorem 1, (AI, +00) is a bifurcation point of Eq. (D). When the functin a(x) changes sign, a result is obtained near the first eigenvalue AI.
°
Theorem 3 Suppose (ad, (ft) hold with < p < q < 1. Then there is a A> Al s.t. for any given A E (Al,A), Eq. (D) has a nontrivial solution, as
In a(x)¢~+l > 0,
(*)
Shaoping Wu
386 For any A E R 1 , a nonhomogeneous case can be delt with:
Theorem 4 Under the conditions of Theorem 3 with f(x,D) =1= D, Eq. (D) has at least one nontrivial solution for any>. =1= Ai, i EN . As >. = Ai the theorem holds if
In a(x)lul q+1 > D,
as '1.1. E Ei \ {D} .
(**)
Finally for a special case of Eq.(D):
-Au = AU -lulq-1u, { ulon = D.
in
n
(D)'
an existence of a changing sign solution is proved in [12). Namely,
Theorem 5
1
As >. > >'2, the Eq. (D)' has a changing sign solution.
The Sketch of the proofs
Let the space E = HJ(n) with the usual norm . Let (" .) and (., .) denote the inner products in spaces HJ (n) and RN respectively. The norms in spaces HJ (n) and U(n) are denoted by 11 ·11 and 11·111' respectively. Define a functional
1('1.1.)
=~
In 1\7'1.1.12 - ~A In 1'1.1.12 +
q
~ 1 In a(x)lul q+1 + In F(x, '1.1.) ,
In
where F(x , '1.1.) = f(x, s)ds . It is routine to show that 1 E Cl and its critical point is a weak solution of the Eq. (D). The Fhkhet derivative of 1 is given by
(i ('1.1.), h) =
In (\7'1.1. , \7h) - >. In uh + In a(x)lulq-1uh + In f(x , u)h.
The inverse operator K of -A in the space E can be given by
(Ku , v)
= Inuv,
'r/u,v E E .
The operator K is selfadjoint and compact in space E. By the spectral theorem,
(Id - AK)u
= E~l (1 -
: )Pju, J
where Pj is the orthogonal projection of the space E on the eigenspace E j
.
A class of resonant or indefinite elliptic problems L e m m a 6 Suppose (a1) and (fl) hold wzth a(%) > 0 on the domazn Then, for any gzven A, there are posztzve constants p and 6 such that I(u)
> 6,
as u
E
E
and 11u//= p
T h e sketch of t h e proof for Lemma 6. As a(x) r 1, the lemma follows from the inequality
where fi > 1 is a fixed number and the constant c = c($) > 0. For general function a(+) > 0 without a positive lower bound, more machineries are needed to evaluate the functional. Suppose X = A, with i > 1. Let the subspace F, = El E2 . . . E, the oi.thogona1 sum of the subspaces. Thus E = F,-1 E, F .: Each element u of E can be decomposed as
+
+ +
+
u = UI
+
+UO
+uz,
where u~ € F , - ~ , u o E E,,uz E F :
It is easy to see that there is a positive constant [ such that ((Id - XiK)u, u)
< -[/lu/)"
((Id - X;K)u, u) = 0,
I
Vu E Fi-1, Vu € E;,
Vu E F:. ((Id - XiK)u,u) 2 [ I / U ~ / ~ , Namely, the quadratic part of the functional is only nonpositive on the finite dimensional subspace Fi. Then it can be controlled by the subquadratic part ' u = 0. The "small sphere" condition is I/UI)$+~ = A q t 1 w a ( z ) l ~ l ' ~ around maintained by the indirect arguments. For details, see [ll]. T h e proof for T h e o r e m 1. It remains to verify the PS condtion. I t can be done by the same idea as above. The boundedness of PS sequence on the : are obtained directly, while obtained by the equivalence spaces Fi-1 and F of the two norms I( . I/ and /I ./I, on the subspace F;. The proof for Corollary 2. It is easy to see that Eq. (D) has no nontrivial solution a t all for A 5 XI. As a consequence of Theorem 1, for each X > X I , there is always a solution uh. By the fact that the constant 6 in lemma 3 is independent of X as X is in a neighbourhood of XI, we can show that / l u ~/(, -t w as A, -t A.: Namely, there is a bifurcation from ( X I , m) for Eq. (D). As the function a(z) changes sign the above methods no longer work. First we give L e m m a 7 Suppose (al) and (fi) hold with p < q < 1. Then the functional I verifies the Palais-Smale condition a3 X # Xi, i E N. And the PS condition holds for X = X i if in addition (**) holds.
Shaoping Wu
388
The sketch of the proof for Lemma 7. It suffices to deal with the resonant case A = Ai . Suppose {un} in E is a PS sequence:
I(u n ) ::; d,
III (un)11 -+ o.
Decompose:
= Unl + UnO + Un2,
Un
where Unl E Fi-I , UnO E Ei,Un2 E Fl·
As in the proof for Lemma 6, there exists a constant 02 > 0 such that
II(Id -
AiK)UnW
>
(A~~l - I)211un1W + (A~:l - I)211un2W
> 02(llunll1 2+ IIUn211 2). So there is a constant 03 > 0 such that 0(1)
= III (un)11 2:: II(Id 2:: 03 (1lunlll + IIun211)
q cllunll P - c c.
AiK)Unll- cllunll - cllunll q - cllunll P -
It gives
IIunll1 2 ::; c(llunll1 2q + IIun211 2q + Ilunol1 2q ) + lower order terms of IIunl1 2q + 0(1) + c, since p < q. On the other hand,
~((Id -
I(un) =
AiK)un, un) + q ~ 1
In
q a(x)lunl +1 +
In
F(x, Un)
Ai ) II I2 1 Ai 2 > 21 ( 1- Al Unl I + 2(1- Ai+l )llun211
j.
- cllunll P+1- cllunll + -1- a(x)lunol q+1 q+1 n + ~I ( a(x)[lunlq+l -lunolq+1j.
q+ in Let A = In a(x)[lunlq+l - Iunolq+lj. By the inequality ab ::;
t + rk =
I, where
IAI <
c
> 0 is an arbitrary constant, we have
f
In
IUno + B[Un2 + unlWlunl + un21
BE [0,1])
< <
cin lunolqlUnl + Un21 + cIn IUnl + un2l q+ Cf
In
q l Iunol +
+ c,
In
IUnl
+ un21 q+l
q < Cfllunoll +l + C,IIUnlllq+l + C,IIU n21I q+l
1
wi
+ cebm
with
A class of resonant or indefinite elliptic problems
389
where t can be taken arbitrarily small. Now the function a(x) may change sign and 11·lla is no longer a norm. For convenience of the expression, we still use it to denote that integral. Claim. Under the assumption of (**) , there is a constant "Y > 0 s.t.
By the claim and the norm equivalence on the eigenspace E i , there is a constant 7r > 0 such that
Thus,
IIUnll~+l =
l
a(x)lunolq+l
+
l
a(x)(lunl'l+l -Iunolq+l)
q
> lIunoll~+l - ctllunollq+l - cEllunlllq+l - cEllun21l +1 > (7r - ct)llunollq+l - CE[lIUnlllq+l + Ilun21I q+l]· Therfore,
I(un) >
>
+ + + Suppose
since 0
Ai )llunlll 2+ 2(11 Ai )11 U n2 112 Al Ai+l cllunll p+l - cllunll + (7r - ct)llunollq+l cE[llunlllq+l + Ilun21l q+l] + lower order terms of IIunl1 2q ~(1- ~: )[lIUnII1 2q + lIu n ol1 2q + Ilun211 2q ] 2q 0(1) - c+ lower order terms of IIunl1 ~(1 - \ Ai )llun211 2- cllunllP+l - cllunll 2 "i+I qI q (7r - ct)llunollq+l - C<[lIUnlll +1 + Ilun21I + ]. 1 2(1-
Ilunll -+ 00 .
< q,p < 1. So
Divide I(u n ) by
Dividing III'(un)11 by
Ilunll, we get
lIunoll -+ 1. Ilunll
Ilunll q+l . Letting n -+ 00,
o ~ 7r -
Ct
we get
> 0,
Shaoping Wu
390
since { can be taken arbitrary small and 0 < p < q < 1. A contradiction. It gives the boundedness of the PS sequence {Un}. From the compactness of the operator K, the sequence {un} has a convergent subsequence. The lemma is proved. The proof for Theorem 3. First we show that the functional is coercive at A = AI . Thus there are constants A > Al and R > 0, such that as Al S A < A,I(u) > 1 for lIull = R. Namely we get the estimate on a big sphere instead of a small sphere. Then by a routine procedure for applying Mountain Pass Lemma, the theorem can be proved. The proof for Theorem 4. Now for f(x,O) :j; 0, we make use of a saddle version of Mountain Pass Lemma [lOJ . It suffices to note that the functional now is coercive on the subspace F/ and -I is coercive on Fi as A E (Ai, Ai+d ensures that I is coercive on subspace F/:_ l and -1 is coercive on Fi - l as A = Ai. Namely there is aM> 0 s.t. I(u) > -M on the space F/( or Fi~l)· Then taking a big sphere SR on the subspace Fi ( or Fi - l ), we have I (u) < - 2M as u E SR . Therefore the values of the functional I on the two sets SR and F/( or Fi~l) are seperated and these two sets are linking. All the conditions for that saddle point theorem are verified and it gives a solution of Eq. (D) . Finally, the term f(x,O) :j; 0 ensures that the solution is non-trivial. Remark 8
For some related work see [15,16j.
The sketch of the proof for Theorem 5. It is easy to see that there is at least one non-positive solution and one non-negative solution for Eq. (D') by Mountain Pass Lemma and the usual truncation technique [6J. In recent years, there are some works on the existence of changing sign solution. See [17,18J and the refernces tchanging sign solution. See [17,18] and the refernces there. Following the ideas in [17,18]' we show that there exists a solution for Eq. (D') outside the set P U (-P) , where P is the nonnegative function cone in the space CJ (0) . However the nonlinear function lulq-lu(O < q < 1) is only Hader continuous now, we have to do some smooth approximations first. Let
where I,
= {q-l . Instead of Eq. (D'), we attack -6.u = AU + leu - g,(u), { ulan = 0,
in
0
Then by the same arguments as in [17], we get a sequence of changing sign solution U, . We show that the sequence is uniformly bounded as { « 1. Moreover, there is a subsequence which is a PS sequence for the original functional .
391
A class of J·esonant or indefinite elliptic problems
In fact we have
~ InIV'U. 12 - ,xu~ + In G(u.)
~ InIV'U.12 - ,xu~ + In G.(u.) + In [G(u.) =
.
+
r
G.(u.)]
(_1__ !)f q+1
Ju <>. q + 1
2
and
In V'u.V'v-,xu.v+g(u.)
(
In V'u. V'v - ,xu.v + g.(u.) + [g(u.) -
=
(
+
r
g.(u.)]
[lu.lq-1u. - fq-1U.]v.
JO$.lu
The later one gives
Thus the sequence u. has a subsequence convergent to a limit u E E, which is a solution of Eq.(D') •. In order to show that it changes sign, we show that not only u. changes sign but also ut and u; has a common lower bound, where u+ = max{u, O} and u- = max{ -u, O}. To multiply the equation -~u.
= ,xu. -
g.(u.)
by ut, we get From the inequality G.(t)
+ cp t p +1
~ ,xt2, '
where G. is the primitive of the function g. and p > 1, we get
= ,x(ut)2 - (q + I)G.(ut) + c., q 1 where c. = (q!l - ~ )f + > 0 and c. -+ 0 as f --+ O. Therefore ,x(Ut)2 - g.(u.)ut
as u. > f,
InIV'UtI2 = In [,x(ut)2 -
(q
+ I)Gl on (Ut)
< ,x(ut)2 - G.(ut)
< c <
1n 1ut1P
+1
cllutIl P+1,
-
c.
Shaoping Wu
392
where c is independent of f . It follows Ilu+1I 2': c. By the same reasoning we have lIu -II 2': c. Therefore there is a measearable set J.L with nonzero measure on which the solution u+(x) =J O,u-(x) =J 0, Vx E J.L. Thus u is a changing sign solution of Eq. (D'). For details, see [12]. Remark 9 The result is true jor more general case
AU - lulq-1u + j(u), = 0,
-~u =
{ ulan where the function jors =J O.
f
in
n
(D")
verifies some additional conditions, jor example, f(s)s
>0
References [1] T. Numba, Density dependent dispersal and spatial distribution of a population, J. Theory Biol.,86(1990), 351-363. [2] H. Brezis and L. Oswald, Remarks on sublinear elliptic equation, Nonlinear Analysis, TMA, 10(1986), 55-64. [3] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems. J. of Func. Anal. 122(1994), 419-519. [4] Liu, Jiaquan and Wu, shaoping, A note on a class os sublinear ellptic problem, Reseach Report, No. 84, 1997. [5] V. Moroz, Solutions of superlinear at zero elliptic equations via Morse theory, Topo. Methods in Nonl. Anal.,10(1997), 387-397. [6] S. Fucik, Nonlinear Differential Equations, Elsevier Scientific publishing Company, 1980. [7] E. Landsman and A. Lazer, Nonlinear purturbation of linear elliptic BVP at resonance, J . Math. Mech. 19(1970),609-623. [8] K. C. Chang and J. Q. Liu, A strong resonance problem, Chinese Ann. Math. 11 B.2(1990), 191-210. [9] Shuji Li and Jiaquan Liu, Computations of critical groups at degenerate critical point and applications to differential equations with resonance, preprint.
A class of resonant or indefinite elliptic problems
393
[10] Ambrosetti,A. and Rabinowitz, P. H., Dual variational methods in critical points theory and applications, J. F\.mct. Anal 14 ( 1973),349-381. [11] Wu, Shaoping and Yang, Haitau, A class of resonant elliptic problem with sublinear nonlinearity at origin and at infinity, to appear in Nonl. Anal. TMA. [12] Wu, Shaoping and Sun, Yijing, A changing sign solution for ellptic equation with sublinear term at origin, preprint, 1999. [13] K. Perera, Multiplicity results for some elliptic problems with concave nonlinearityies, J . of Diff. Eqs. 140(1997),123-133. [14] M. Romos , Remarks on resonance problem with unbounded perturbations, Diff. and Integ. Eqs. 6:1(1993), 215-223. [15] Liu, Jiaquan, Sign-changing solution fot elliptic equations of second order, preprint, 1999. [16] T. Bartsch and Z.Q . Wang, On the existence of sign changing solutions for semilinear Dirichlet problems, Topological methods in nonlinear analysis,7(1996), 13-131.
Geometric Structures of Solutions for Certain Elliptic Differential Equations Xue-Feng Yang Department of Mathematics & Statistics McMaster University Hamilton, On, Canada L8S 4Kl E-mail: yangxtDicarus.math.mcmaster . ca
In this article, I will give a summary of my research on the global qualitative behaviors of solutions of certain elliptic ordinary differential equations, partial differential equations and variational problems. Some comments, observations and open problems will be also discussed .
1
Inverse nodal problems
In 1830's, Sturm and Liouville published a series papers on a general boundaryvalue problem
-y" + q(x)y
= >..y,
+ y' (0) sin Q = 0, y(l) cos{3 + y'(l) sin{3 = O.
y(O) cos Q {
(1.1.1)
The first part of their theory described the nature of the spectrum, which was the motivation of the Fredholm's study of integral equations and for the Hilbert-Schmidt method. It is of importance in quantum mechanics and the operator theory today. The other part des cribs the nodes of the eigenfunctions.
394
Geometric structures of solutions
395
For each n ~ 2, the n-th eigenfunction has n - 1 nodal points so that 0< x;(q,a,{3)
< x~(q,a,{3) < ... < x~-l(q,a,{3) < 1.
In the theory, the nodes X(q,a,{3) = {x~(q,a,{3) : i = 1, ',n -1,&n = 2, . . . }
are visible as those of general functions on [0,1], and eigenvalues are audible, for example, in string models. Sturm-Liouville theory is the bridge between what people can hear and what they can see, which helps people to understand the geometrical structure of the solutions. In 1950's, Gel'fand and Levitan[9] started to consider the inverse spectral problem, which they were interested in how much physical information can be heard from its sounds. It is also nice to know how much physical information can be seen from its nodes. In 1988, McLaughlin [16] started to study such a problem, which is so-called the invers nodal problems. I will focus on the inverse nodal pJ.'Qblem in this section.
1.1
What is the inverse nodal problem?
Working for an inverse nodal problem, people are seeking a way to answer whether and how the potential and the boundary data are determined from the nodes of the eigenfunctions. The following uniqueness result was proved by McLaughlin [16] the case of a = {3 = 0, by Hald-McLaughlin [11] the case of a :f. 0 and {3 :f. 0, and Yang [18] the rest of the cases. Theorem 1.1. Assume that q,ij E L1[0,1] and 0 ::; a,{3,ti,{J
<
7['.
If there
exists a subset A C {(j,n)lj = 1,,, , ,n -1,n
= 2, .. ·}
so that cl{x~(q,a,{3)I(j,n)EA}=[O,I], and
(1.1.2)
x~(q,a,{3) = x~(ij,ti,{J) , (j , n) E A,
(1.1.3)
where cl(B) is the closure of B under the natural topology on [0,1], then (q
_11
q(t)dt, a,
(3) =
(ij -
11
ij(t)dt, ti,
(J) .
396
Xue-Feng Yang
1.2
An Explicit Formula
McLaughlin's result [16] actually initiated the study of the inverse nodal problem. Based on this theorem, I raised the following questions in the paper [18].
(1) (Existence) When is a set Y = {y~lj = 1,2, ,,, ,n - 1; n = 2,3, .. ·} C [0,1] a nodal set? In other words, for what Y does there exist q E Ll[O,I],a,,B E [0,11') such that Y = X(q,a,,B)? (2) (Determination) Does there exist an explicit formula for q, a,,B in terms of the nodes? Although problem (1) is still open, it is the first time that an explicit formula
(q,a,,B)
= F({x~})
for this specified problem has been offered, which is a solution for problem (2). Theorem 1.2. Assume that X = {x~lj = 1,,,, ,n -1; n = 2,,,, } E NS[O,I]. Then one and only one of the following limits exists
(1.1.4) (1.1.5)
(n - ~) x~ - j] (n - ~) ,
. lim
[
. lim
[nx~ -
~-+x , ~# :--+x,:-#x
2
j] n,
2
(1.1.6) (1.1. 7)
for all x E [0,1]. Denote by the limit ~g(x),x E [0,1]. Then g(x) is absolutely continuous and d~g(x) exists almost everywhere.
Geometric structures of solutions
397
Furthermore, if taking
a = {-cot- 1 (g(0)),
for cases (1.1.6), (1.1.7),
0,
(J - {- cot- 1 (g(1)),
then X(q,a,(J)
=X
for cases (1.1.4), (1.1 .6), for cases (1.1.5),(1 .1.7),
0,
q(x) = 2
for cases (1.1 ·4), (1.1.5),
(d~g(X) + g(O) -
and
9(1)) ,
f; q(y)dy = O.
The formula involves the limit (1.1.4), (1.1.5), (1.1.6), and (1.1.7), which may seem initially be a defect. The point is that X has only a countable number of points whereas q is defined on interval [0,1]' which is uncountable. In order to clarify this point, let us compare the result with Taylor'S series. Suppose f is a real analytic function in (-f,f). If one knows {J(n)(o)}~=o, which is a countable set, then one can reconstruct f(x),x E (-f,f), from Taylor'S series
f( x) -
L
oo
n=O
f(n) (0) n _ l' X-1m n! k-+oo
Lk n=O
f(n) (0) n x. n!
It is impossible, however, to reconstruct f(x) without using a limit.
1.3
Overdetermination and a New Inverse Nodal Problem
From the above uniqueness theorem, the inverse nodal problem is overdetermined. It is natural to ask if nodal information on some subinterval of [0,1] can determine the potental and boundary conditions. In this subsection, some results will be shown for this purpose. Let
T = {(j,n)lj = 1,,,, ,n -l,n For 0 <
= 2, ... }.
b::; 1, and ACT = {(j,n)lj
= 1,,,,
,n -l,n = 2, .. ·},
Xue-Feng Yang
398
define B(A, b, q, a, (3)
{x~(q,a,(3) E (O , b)l(j,n) E A , j = 1, .. · ,n -I,n = 2, ··· }, B(n, A, b, q, a, (3) {x~(q,a,(3) E (O,b)l(j,n) E A,j = 1,··· ,n -I}.
Notations B(A) and B(n, A) will be used if there is no confusion for the about two sets. B(A) is the set of the nodal points in (0, b) with their indices in A, and B(n, A) is the set of the nodal points in (0, b) of the nth eigenvalue with their indices in A . Definition 1.1. Let {nk : k = 1,2, ·· ·} be a subset of the positive integer set such that
B(A), ACT, is called twin in [0, b) about (p, a, (3) in the Sturm-Liouville problem (1.1.1) if for any k ~ 1, B(nk,A) contains a pair of adjacent nodal points X~kk(q,a,(3) and X~kk+1(q,a,(3) . B(A), ACT, is called s-dense in [0, b) if there is a subset of {nd (also denote by {nd) such that for any x E [0, b) there is X~k E B(nk,A) with lim x~ = x .
k-+ oo
k
B(A), ACT, is called dense in [0, b) if
d(B(A))
= [O,b) .
The following two theorems have been proved in [20). Theorem 1.3. Let ~ < b ::; 1 and 0 < c: and s-dense in [0, b) about (q , a,(3), and
< 2b -
1. Assume that B(A) is twin
#{nk: nk::; n} ~ (1-c:)n+ for sufficiently large integer n
3c:
2'
(1.1.8)
> o. If there are Jk and nk such that (1.1.9)
Geometric structures of solutions
399
for all integer j 's satisfying jk
+j
E
A
then
Theorem 1.4. Let! < b::; 1 and 0 and dense in [O,b] about (q,a,{3),
< E: < 2b - 1. Assume that B(A) is twin
a=ct,
and 3E:
# {nk : nk ::; n} ::::: (1 - E:)n + 2' for sufficiently large integer n > O. If there are lk and fik such that
for all integer j 's satisfying jk
+j
E A,
l(nk,A,b,q,a,{3) ::; jk l(fik' b, ij, ct,~) ::; lk
+ j::; r(nk,A,b,q,a,{3),
+j
::; r(fik, b, ij, ct, fi),
then
It is very surprising that the nodes in a subinterval of the domain determine the Sturm-Liouville problem completely. Back to the uniqueness(see Theorem 1.1) of the inverse nodal problem, it is difficult to obtain the exact subscript and superscript of a nodal point without knowing all of the nodal points in (0,1). This is actually a global condition. So when only knowing the partial information of the nodal set, it is not reasonable to assume such sort of the condition as (1.1.3), which assumes not only the same subscript, but also the
Xue-Feng Yang
400
same superscript on both sides of the equality. Instead, the condition (1.1.9) requires only that all of pairs of nodal points in two Sturm-Liouville problems are same if they are in the certain subinterval and have the certain order. Although this condition makes the estimate of the difference of the eigenvalues in two Sturm-Liouville problems much harder, it is more reasonable from both mathematical and physical point view. For this reason, the result is new even if b = 1. For the inverse spectral problem, Hochstadt and Lieberman [13] showed in 1978 that the potential is uniquely determined by the boundary condition, the spectrum(necessarily simple), and the potential on [0, Some refinements of the Hochstadt and Lieberman's result are presented by several authors(see [10] and the references therein). It is worth to point out that the subinterval [0, ~] cannot be replaced by any subinterval [0, ~ - c], c > O(see [10]) in their result. Comparing with the other data in Hochstadt and Lieberman's result [13], nodal points determine uniquely the potential as well as its boundary condition of the Sturm-Liouville problem. It remains open if the result holds true for b E (0, It is also an open problem if my result is valid provided the condition holds in any non-empty open subinterval. I call the later a gene property. Condition (1.1.8) is motivated by (1.1.10) in the following Gesztesy and Simon's inverse spectral theory, but the statement of (1.1.8) is independent of the spectrum.
n
n
Proposition 1.1. There is Ao E IR such that the potential q a.e. 2n [0, ~ + for some b E (0, 1), Q E [0,71"), and a subset {nd of the integer numbers satisfying
!]
(1.1.10) for all sufficiently large A ~ Ao, uniquely determine (3 E [0,71') and q a.e. on [0,1]. Since only partial data on nodal sets are known, it is difficult to recover all of the eigenvalues. Thus, results like Proposition 1.1 are crucial to the uniqueness of the new inverse nodal problem. The strategy of the proof in this paper is to convert the new inverse nodal problem into the inverse spectral problem of Gesztesy and Simon's [10], based on some estimates on eigenvalues and potentials. It seems to the author that
Geometric structures of solutions
401
the result in the present work is the first one which bridges the inverse nodal problem to the inverse spectral problem.
1.4
Why the Existence Problem is So Hard?
I will give some remarks to concern the existence of the inverse nodal problem. These remarks are not new, but this is the first time to publish them. I gave such remarks in several talks. Here I will not talk any possiblity to solve this problem, but will try to show the problem is very difficult from conventional mathematics. These remarks come from the uniqueness result and the explicit formula in above two subsections. From Theorem 1.1, the inverse nodal problem is overdetermined. Actually, for any proper subset A of the nodes, if it is a dense subset of [0,1], then X(q, a, (3) \ A is determined by A. Let us look at this example closely. Define
A
= X(q,a,f3) \
{x~(q,a,f3)} .
So xHq, a, (3) is uniquely determined by A . This implies for any does not exist qo E Ll[O, 1] and ao,f3o E [O,~) such that
X(qo, ao, f3o)
€
:f:. 0, there
= Au {x~(q, a, (3) + €} .
We also can construct non-existence examples when A is the dense complement set of any subset of X(q,a,f3) . This shows us that it is difficult to perturb the nodes. If we recall various existence theories in analysis, for instance, the contraction mapping principle, the degree theory, the critical point theory, and so on, most of them remain true under small pertubation in some sense. On the other hand, the above example shows us that there is no solution for the inverse nodal problem under very simple and natural perturbation. TUrn to the remark on the explicit formula. For simplicity, assume that the limit [nx~ -
. lim
j] n
*"--+x , *"#x
exists for all x E [0,1]. Define a function Q : [0, 1]
Q(x)
~{
[nx~ ~g(x),
j] n,
if x =
otherwise,
ft,
~
1R:
and (n,j) = 1,
Xue-Feng Yang
402
where (n,j) is the greatest common factor of nand j. It is easy to see that
R(x) = 0,
lim "'-+"'0,"'#"'0
where the remainder function
R(x) = [Q(x) -
:2 g(X)] .
It is worth to point out that remainder function R(x) has the same property as Riemannian function
Rieman(x) = {
*'
0,
if x = and (n,j) otherwise.
= 1,
Hence for any given nodal set, One can constructs a function Q from the nodes. FUrthermore this function is the sum of an absolutely continuous function and a function of Riemannian type. In the case of q = 0 and a = (3 = 0,
R(x)
= 0, x E [0,1].
I don't know if there is an example, whose remainder function is non-zero. If there is no such a example, then Q itself is absolutely continuous. So the existence problem is solved since in this case the potential and the nodes can be solved directly from the function Q. I guess that there exists such an example which is not zero. If there is such an example, then there are two issues which we have to solve. The one is if for any absolute continuous function g, there is X (q, a, {3} and the function Q such that lim
"'-+"'0,"'#"'0
Q(x)
= 1C'-\g(x).
Unfortunately, this is not true. Then what is the set of all possible absolute continuous functions such that the above limits hold? Is this set a closed set under the certain topology? The other is how to check the condition on the remainder function if assuming we can solve the first issue. It is really a hard problem to find a remainder function in all functions of Riemannian type. This family of functions is not closed under the max-norm topology, so again it is hard to apply functional analysis to approach the problem. So far, the method we used is the asymptotic analysis about the nodes and eigenvalues. By use of the asymptotic analysis, at most, we can say some
Ueometric structures of solutions
403
structure results, such as, the nodes must in certain class, which is from an absolute continuous function plus a function of Riemannian type. This actually is a necessary condition. It is easy to see that it is impossible to answer any question in the above discussion in this way. We have to develop new idea for this purpose.
2
Sturm-Liouville Theory for Super-linear PDEs
Another theme is the attempt to understand the qualitative properties of the solutions for the super-linear elliptic partial differential eqations. Nodal sets of partial differential eqations, unlike ordindary differential eqations, are far from being well understood. A type of Sturm-Liouville theory for super-linear ordinary differential equations was obtained by Coffman [5], in which he drew on Lyusternik-Schnirelman theory and Morse theory, but for super-linear partial differential equations, people will see that it is much more complicated and difficult. A totally different method is applied here.
2.1
Two Types of the Generaliztions
Several people have tried to generalize the Sturm-Liouville theory to the elliptic problem
{
-6U
= f(x,u)
Ul8M
= 0,
in
M,
where M is an N-dimensional closed Riemannian manifold or compact Riemannian manifold with a smooth boundary. Possibly the first result in this area is the Courant Nodal Domain Theorem [6], which is a topological version of the Sturmian theorem on counting nodes. Courant's theorem states that the number of the nodal domains of Un is less than or equal to n, where a nodal domain is defined as a connected component of M\u- 1 (0). Kreith [15] generalized the Courant nodal domain theorem to the following nonlinear elliptic problems
{
-
E~j=l 8~i (aij(x) :~) + c(x, u, \1u) = 0, in M, ~~ '-1 aiJ' 88u. cos(v, Xj) + s(x)u = 0, on 8M, L.J't,Jx,
Xue-Feng Yang
404
by use of a generalized Picone identity. His result is that the number of the nodal domains of a solution u is less than or equal to the number of the nonpositive eigenvalues of the linearized equation at u . Later Bahri-Lions [2] and Benci-Fortunato [3] obtained better estimates for the super-linear elliptic equations with Dirichlet boundary conditions, finding that the number of nodal domains of a solution u is less than or equal to the Morse index, which we will define later. Their proof, based on the variational principle, is simpler and elegant. On the other hand, Courant himself found some examples where the n-th eigenfunction has only two nodal domains for large n. For manifolds with analytic metrics, Donnelly and Fefferman [7] [8] found bounds for the Hausdorff measure of the nodal set of the n-th eigenfunction of the Laplace-Beltrami operator in terms of the n-th eigenvalue, which is an analytic generalization of Sturm-Liouville theory. The bound was two sides:
where G l , G2 > O. It is still open whether the same is true assuming only that metric is smooth(see Yau [22]).
2.2
Nodal Set and Mores Index
In the article [19] I develop a type of Sturm-Liouville theory for certain superlinear elliptic PDE's as follows. Assume that 0 is a domain in JRN, and 6. is the standard Laplacian on O. For super-linear equations, under certain conditions it is known that there are infinitely many solutions with both Loo-norm and Morse index unbounded (see [1], also see [4] and references therein) . For general super-linear elliptic PDE's, after developing a new boundary estimate, I obtain the following estimate of the LOO norm of the solution via its Morse index:
IluIILOO(f1) :s G(1 + ind(u))"' , where ind( u) is the Morse index of the solution u. This result not only answers a question posed in Bahri-Lions [2], but also provides an explicit bound. Furthermore, using this result, as well as the Jerison and Kenig's Carleman-type inequality [14], and an iteration method from Donnelly-Fefferman [7], an estimate of the vanishing order r(x, u) ofthe solution is obtained of the following
Geometric structures of solutions
405
form: T(X, U) ::; 2C(1+ in d(u))"' .
Examples are known which indicate that one could not use only a purely local approach to prove the above estimate. Using my £,>0 estimate, as well as the Jerison and Kenig's Carleman-type inequality [14], and Hardt-Simon's estimates [12], we obtain an estimate of the N - 1 Hausdorff measure of the nodal set of the solution:
These are the first global estimates on the vanishing order and the measure for nonlinear partial differential equations. In this case, the left sides of these inequalities are the observable quantities, and the right sides are the functions of the Morse indices, which are defined for certain functionals over Sobolev spaces.
2.3
Singular Perturbations on Non-convex Variations
Consider the following singular perturbation problem Je(u)
= ~ In £/Vu/ 2 + ~W(U)dX, u E Hl(O) ,
where W E C(JR, JR+) has exactly two zeros a < (3, and 0 C JR2 is a bounded open region with smooth boundary. Je is the energy functional from the context of the Van der Waals-Cahn-Hilliard theory of phase transitions. Modica [17) studied the minimizers with the constraint on the density. It is natural to study the existence of non-minimal solutions and their geometric nature as a singular perturbation, which is a possible way to explore the multi-layer phenomenon. In a recent work [21), I started to consider the existence of non-minimal solutions and their geometric nature for the above functional. The functional has two unique global minimizers, u = a and (3. Since the functional may not be smooth and it is non-convex, so critical point theory of Ambrossetti and Rabinowitz [1) (also see [4)) cannot be applied here, but a minimax value of the mountain pass type can be defined as follows
CE = inf max JE(h(t)), hH099
Xue-Feng Yang
406
where r is the set of all continuous paths in Hl (0) joining Q and (3. "From the construction, for sufficiently small c:, d > 0, there are a path h£.6 E rand t£.6 E [0.1), such that
h£.6 is called to be an approximating mountain pass, and U£.6 = h£.6(t£.6) an approximating mountain pass point. Based on a fine analysis on the classical Dido's problem, I showed that the following result[21): If the domain is symmetric about origin, then for any set of approximating moutain passes, there exist approximating moutain pass points such that the interface of the limiie·~'l~~6blem contains finitely many arcs in 0 whose both -. 1 1 ~'" ends are in ao. ' It is worth to mention that if 0 is an ellipse, then the interface is its short diameter. Some nonsymmetric variants are also discussed in the paper [21].
2.4
Some Comments
The existence of the nodal point has not been discussed in this paper. Fortunately, some papers can been found in this proceeding, which are about the topic of the sign changing solution. This is one important way to approach this problem. The estimate of the lower bound of N-1 dimensional Hausdorff measure of the nodal set is still open. In the perturbation problem, we have a better understanding on the nodal set as discussed above. The result depends on the nonlinearity and the domain. There are some other examples which people can be found in this proceeding.
References [1] Ambrosetti, A. and Rabinowitz, P. H. Dual variational methods in critical point theory and applications. J. Funct. Anal., vol. 14, 369-381(1973). [2] Bahri, A. and Lions, P. L. Solutions of superlinear elliptic equations and their Morse indices. Comm. Pure Appl. Math., vol. XLV, 1205-1215 (1992).
Geometric structures of solutions
407
[3] Benci, V. and Fortunato, D. A remark on the nodal regions of the solutions of some superlinear elliptic equations. Proc. Royal Soc. Edinburgh, lIlA, 123-128 (1989). [4] Chang, K. C. Infinite Dimensional Morse Theory and Multiple Solution Problrms. Birkhiiuser, PNDLE vol 6, (1991) [5] Coffman, C. V. Lyusternik-Schnirelman theory:complementary priciples and the Morse index. Nonlinear analysis, theory, methods and appl., vol.12, 506-529(1988). [6] Courant, R and Hilbert, D. Methoden der mathematischen physik. vol. 1, Springer, Berlin (1931). [7] Donnelly, H. and Fefferman, C. Nodal sets of eigenfunctions on Riemannian manifolds. Invent. Math., vol 93, 161-183 (1988) . [8] Donnelly, H. and Fefferman, C. Nodal sets of eigenfunctions: Riemannian manifolds with boundary. Analysis, Et Cetera, Academic Press, Boston, MA, 251-262 (1990). [9] Gel'fand, 1. M. and Levitan, B. M., On the determination of a differential equation from its spectrum, Izv. Akad. Nauk SSSR Ser. Mat., vol. 15, 309-360 (1951). [10] Gesztesy, F., and Simon, B. Inverse spectral analysis with partial information on the potential, II. The case of discrete spectrum. to appear in Trans. Amer. Math. Soc.(1999) [11] Hald, O.H. and McLaughlin,J.R, Solutions of inverse nodal problems. Inverse Problems, vol. 5, 307-347 (1989) [12] Hardt, R and Simon, L. Nodal sets for solutions of elliptic equations. J. Diff. Geom., vol. 30, 505-522(1989). [13] Hochstadt, H., and Lieberman, B. An inverse Sturm-Liouville problem with mixed given data. SIAM J. App. Math., 34, 676-680 (1978) [14] Jerison, D. and Kenig, C., Unique continuation and absence of positive eigenvalues for Schrodinger operators, Ann. of Math., 121, 463-494(1985). [15] Kreith, K. Nodal domain theorems for general elliptic equations. Rocky Mountain J . of Math ., vol. 1,419-425 (1971). [16] McLaughlin, J. R, Inverse spectral theory using nodal points as data-a unique result, J. DifJ. Equ., vol. 73, 354-362(1988).
408
Xue-Feng Yang
[17] Modica, L. Gradient theory of phase transitions and minimal interface, Arch. Rational Mech. Anal., vol. 98, No.2, 123-142(1987). [18] Yang, X. F . A solution of the inverse nodal problem. Inverse Problems, vol. 13, 203-213 (1997). [19] Yang, X. F. Nodal sets and Morse indices of solutions of super-linear elliptic PDE's. J. Functional Analysis, 160, 223-253(1999) . [20] Yang, X. F . A new inverse nodal problem. J. Differential Equations, a special edition dedicated to Hale's birthday(2000). [21] Yang, X. F., Non-minimal solutions of a singular perturbation on nonconvex variational problems: I, preprint (1999). [22] Yau, S.T. Problem section, Seminar on Differential Geometry. Princeton University Press, (1982)
KAM Theory for Lower Dimensional Tori of Nearly Integrable Hamiltonian Systems Jiangong You Department of Mathematics, Nanjing University Nanjing 210093, P.R.China E-mail: [email protected]
Abstract. In this article we give a brief survey for the recent developments in the KAM theory for lower dimensional tori of nearly integrable Hamiltonian systems and their applications in the study of invariant manifolds in the resonant zone and the construction of quasi-periodic solutions of Hamiltonian partial differential equations. The dynamics of the integrable Hamiltonian systems is simple in the sense that all the compact energy surface are foliated by invariant tori which carries quasiperiodic motions of the corresponding Hamiltonian equations. But integrable Hamiltonian systems are rather rare in the whole family of Hamiltonian systems. One of the landmarks in dynamical systems, especially in Hamiltonian dynamical systems, is the KAM (Kolmogorov-Arnold-Moser) theory, which tells us that for all Hamiltonian systems in a open neighborhood of a nondegenerate integrable Hamiltonian systems, the quasi-periodic motions in invariant tori are typical (See [Arnold, Encyclopedia] and the references therein). Later [Melnikov, 1965] formulated a KAM type persistence result for lower dimensional tori of nearly integrable Hamiltonian systems. In recent years, the KAM theorem for lower dimensional tori attracts more attentions for its applications in the construction of quasi-periodic solutions of Hamiltonian partial differential equations and the study of the dynamics in the resonant zone of nearly integrable systems.
1
Persistence of lower dimensional tori
We start with a real analytic Hamiltonian
H(x, y, u) = h(y, u)
409
+ P(x, y, u)
(1.1)
Jiangong You
410
in (x, y, u) E Tn X D X R 2m C Tn X Rn X R 2m with the symplectic structure 2:~=1 dXi /\ dYi + 2::'1 dUi /\ du_;. The corresponding Hamiltonian equations are dy - = -Px, dt where J is the standard symplectic matrix in R2m. Assume that hu(Y,O) = O,dethuu(Y,O)"# 0, and P is small, i.e., the unperturbed Hamiltonian system defined by h possesses an invariant sub-manifold x E Tn, y ED, u = 0 foliated by a family of invariant tori x E Tn, y = Yo, u = 0 and the flow on each torus is given by x(t) = Xo + hy(Yo, O)t, Y = Yo, u = O. If P is not zero, the sub-manifold x E Tn , y E D, u = 0 is no longer invariant. By KAM theory, one could expect that a Cantorian submanifold with positive Lebesgue measure in x E Tn, y ED, u = 0 persist the perturbation. Expanding the Hamiltonian (1.1) in the neighborhood of u = 0, we have
H(x, y, u)
= h(y, 0) + ~(huu(Y, O)u, u) + P(x, y, u) + O(u 3).
For the unperturbed system, the local normal behavior of the invariant torus y = Yo, u = 0 is determined by the matrix huu(Yo, 0) if it is non-degenerate. Linearizing the Hamiltonian in the neighborhood of torus Tn X {y = ~ E D} X {u = O}, we arrive at a family of perturbed integrable Hamiltonians,
H
1 = N + P = (w(~), y) + 2(A(~)u, u) + P,
(1.2)
where (x,y,u) E Tn X Rn X R 2m, w(~) = hy(~,O),A(~) = huu(~,O), and P = P + 02(y) + O(yu) + 03(U). We shall treat ~ E D as an independent parameter. This reduction reduces the persistence problem of invariant tori of a fixed Hamiltonian system into the persistence problem of an invariant torus of a family of perturbed linear Hamiltonian systems. The setting has been frequently used by many authors. In this paper, instead of (1.1), we shall formulate all results for (1.2) with an independent parameter ~ varying over a positive measure set D. For simplicity, we assume that H is analytic in all variables including ~ . If the unperturbed torus is elliptic, i.e., all the eigenvalues of JA(~) are pure imaginary and simple, (1.2) can be transformed to the following Hamiltonian n
H
1
m
= N + P = LWj(~)Yj + 2 L j=1
j=1
nj(~)(u; + U~j) + P,
(1.3)
KAM theory for lower dimensional tori
411
by a linear symplectic coordinator transformation. Melnikov proposed the following non-resonant conditions to guarantee the persistence of lower dimensional tori: (k,w(~))
+ ni(~)
:i 0,
(1.4)
+ ni(~) + nj(~) :i 0, (k,w(~)) + ni(~) - nj(~) :i 0, Ikl + Ii - jl ::f: 0, (k,w(~))
(1.5) (1.6)
where w = (WI,'" ,wn ). For the persistence result, we only need to consider (1.2) or (1.3) in the complex domain D(r,s)
= {(x,y,u)1
IIImxl
Theorem 1 ([Melnikov, 1965], [Eliasson, 1988]) Suppose that (1.3) is real analytic and satisfies {L4}, {1.5}, {1 .6}. Then for a given 7 there is a small constant 10 depending on n, T, r, L, 7, such that if the vector field Xp of P in D(r, s) satisfies IXpID(r,s)xD
~
10,
the following conclusions hold true: there exists a Cantor set D"( CD, a Whitney-smooth map
and a diffeomorphism w: D"( -+ Rn, such that the map ~ restricted to Tn X {w } is a real analytic invariant torus with frequencies w '" W for the Hamiltonian {1.3} at w. Moreover
ID -
D"(I -+ 0, as 7 -+ 0.
This result was announced in 1965 by Melnikov[19]. The complete proof was carried out many years later by [Eliasson, 1988]. Here the formulation is slightly different from their original papers, but the difference is not essential. Some infinite dimensional versions of this result were given also by [Kuksin, 1988] and [Poschel, 1989]. We remark that, in this case, only the measure estimate is available. One does not know if (1.3) has a torus with prescribed frequencies. Now (1.4) is generally referred as the first Melnikov's non-resonant conditions, while (1.5) and (1.6)are referred as the second Melnikov's non-resonant conditions. If all eigenvalues of J A are away from the imaginary axis, the torus is called hyperbolic. The hyperbolic lower dimensional torus seems more robust
Jiangong You
412
than the elliptic lower dimensional tori. In this case, for any given ~ E D if w(O = (Wl(O, '" .wn(O) satisfying the Diophantine condition for
T
> n - 1, 0 "I- k
E
zn,
(1.7)
we have the following theorem Theorem 2 (Moser[20], Graff[12)) Suppose that w(~) is non-degenerated, i.e., det ~e "I- O. Then for any given w(O satisfying (1.7), if the perturbation is analytic and sufficiently small, there is a C close to ~ such that (1.2) at C possesses an invariant torus with prescribed rotation frequencies w(O·
Note that Theorem 1 for the elliptic case, where the second Melnikov's conditions is assumed, excludes the case of multiple normal frequency, i.e., Oi = OJ for some i "I- j . While Theorem 2 for the hyperbolic case need not the second Melnikov's non-resonant condition (the first Melnikov's non-resonant conditions are automatically satisfied by the hyperbolicity). It is a nature question to ask if the second Melnikov's conditions are necessary for the elliptic case. Meanwhile, the multiplicity of normal frequencies is essential in some important applications such as Hamiltonian PDEs with periodic boundary conditions and lower dimensional tori in the resonant zone. With the above motivations, in recent years there are some progress towards releasing the second Melnikov's conditions. Theorem 3 (You[31], 1999) If the eigenvalues of A(w) satisfy (1.4), (1.5), (1 .6) with k =f. 0 replacing Ikl + Ii - jl "I- 0, and the perturbation is analytic and small, the conclusions of Theorem 1 hold for Hamiltonian (1 .2).
The main significance of replacing Ikl + Ii - jl "I- 0 by Ikl "I- 0 in Theorem 3 is it allows the multiple normal frequencies . Meanwhile, Theorem 3 is given for more general Hamiltonian (1.2) rather than (1.3) since the normal form (1.3) is not always available in case of normal multiplicity. Some partial results for multiple normal frequency case was also obtained by [Xu, 1997]. Recently, Xu and You[29] gave a KAM theorem of lower dimensional tori under the first Melnikov's condition. Theorem 4 (Xu and You[29], 1999) If (w(~), O(~)) satisfy the first Melnikov's conditions (1·4) and the perturbation is analytic and small, the conclusions of Theorem 1 hold for Hamiltonian (1.3) .
Proofs for all the above results follows in principle the standard KAM method, i.e., reducing the perturbed Hamiltonian system to a norm form with
KAM theory for lower dimensional tori
413
an invariant torus by infinite many coordinator transformations. To here, we have to mention the Bourgain's remarkable result on the Melnikov's persistence problem. Theorem 5 (Bourgain[21,1997), if w, n in Hamiltonian (1.3) satisfy the first Melnikov's conditions (1·4) and P is analytic and small, then for f.L "" 1 taken in a set of positive measure, (1 .3) possesses a perturbed torus with frequency f.LW, parametrized as y
= y(t),
,x
= f.LWt + x(t),
u
= u(t)
with y(t), x(t), u(t) quasi-periodic with frequency vector f.LW.
Bourgain's result is given for the constant normal frequency case, which can also be proved by the method in Theorem 3. But we believe that Bourgain's approach works for the general parameter dependent case and leads to the general conclusions of Theorem 4. Here we briefly compare the approaches of Theorem 4 and Theorem 5. The results in two theorems are more or less same. For the proofs, both of them use infinite many coordinator transformations to reduce the perturbed Hamiltonians to a normal form (w, y) + (A(x)u, u) which has an invariant torus. The difference is: in Bourgain's approach, he kept all the x-dependent second order terms unsolved and waved them to the normal form part, which surely makes the homological equations in KAM iterations more complicated. As a result, A multiscale analysis is needed to control the inverse of a linear operator. By this approach, the second Melnikov's nonresonant conditions are avoided. In Xu and You's approach, they observed that, after disregarding a small set of parameter, most of non-resonant relations in the second Melnikov's conditions are satisfied automatically although we only assume the first Melnikov's conditions (For example, if all the normal frequencies are constant (independent of ~), the second Melnikov's conditions are satisfied except a small set of parameters) . This observation makes it possible to solve most of the x-dependent second order terms of the form aij (x )UiUj in the homological equations. Only the terms corresponding to the resonance (k, w) + 2n i = 0 are waved to the normal form part. As a result, the normal form part is simpler and then the homological equations is simpler in the next KAM iteration step. In fact, the homological equations can be reduced to some element finite dimensional linear algebraic equations with an uniform dimension bound at each KAM step. This avoids the multi scale analysis for bounding the corresponding green's functions. Xu and You's approach, in some case, is quite simple and yields somewhat stronger results ( x-independent normal forms, linearly stability, explicit Diophantine properties of the obtained Tori). For example, we have the following
Jiangong You
414
Theorem 6 (Xu and You[28]) Ifw({), O({) in (1 .3) satisfy (k,w({)}+Oi({) :f. 0, and (k,w({)} + 20 i ({) :f. 0, and the perturbation is analytic and small, the conclusions of Theorem 1 hold for Hamiltonian {1 .3}. The proof of Theorem 6 is given in (28). The main idea is to employ a symplectic change of variables which reduces the problem almost to the multiple normal frequency case, then the proof of Theorem 3 works for this case. If the first Melnikov 's conditions are not satisfied, the lower dimensional tori might be destroyed by arbitrary small perturbations. Actually, this case can be reduced to the zero normal frequency case by a nonlinear symplectic change of variables, thus there is no persistence result ( see examples in (31), [28]) . However, with some higher order terms in the normal form, the persistence result is also available. In (31), we considered the following simplest case:
1 2 -u2d +Px,y, ( ) H=(w,y)+2V u,v,w,
(1.8)
where (x,y , u,v) E Tn X Rn X Rl X Rl and d ~ 2. Note that, in (1.8), the normal space is 2-dimensional and the normal frequency is zero.
Theorem 7 (You[30]) For any given w satisfying {1.7}, if the perturbation is analytic and sufficiently small, there is a w* close to w such that (1.8) at w* possesses a normally (weak) hyperbolic invariant torus with prescribed rotation frequencies w. The proof of Theorem 7 is quite involved, we do not know if it can be generalized to the multiple zero normal frequency case.
2
Lower dimensional tori in the resonant zone
Consider a Hamiltonian vector field X H with the associated Hamiltonian H(x,y)
= h(y) +€P(x, y),
(2.1)
where y E D C Rn, x E Tn , hand P are real analytic functions defined on a complex neighborhood of a bounded domain D and the torus Tn(= Rn /27rzn), respectively; h satisfies the standard nondegeneracy condition detf.4(y) :f. 0 in D ; P is a perturbation and € > 0 is a small parameter. Y For the unperturbed Hamiltonian h(y), w = ~Z (Yo) is called nonresonant, if it satisfies (k , w) :f. 0 for any k E zn \ O. Otherwise it is called resonant. w is called a multiplicity m resonant frequency if there is a rank m subgroup g of zn generated by independent integer vectors 1'1, ' .. ,1'm such that (k, w) = 0 for all kEg and (k,w) :f. 0 for all k E zn/g .
415
KAM theory for lower dimensional tori
For any given m-dimensional subgroup 9 of O(g,D)
zn,
= {y ED: (k,w(y)) = O,k E g}
is a mo = n - m dimensional surface, which is called a g-resonant surface. Locally it is diffeomorphic to Rn-m. Since KAM theory is a local theory, we may assume that O(g, D) is globally diffeomorphic to a sub domain in R n - m without loss of generality. For the trivial subgroup 9 = 0, according to the celebrated KAM theory(see [1] and the references therein), most of the nonresonant tori of the integrable system persist small perturbations. For a given O(g, D) with dimg = m > 0, Tn X O(g, D) is a resonant invariant manifold of X h , in which all the ndimensional invariant tori are foliated by mo = n - m dimensional lower dimensional tori. What happens to the resonant torus of the unperturbed system with frequency aa hy (y) for y E O(g, D) under a small perturbation? In general, it will . be destroyed by the perturbation. Note that if y E O(g, D), the n-dimensional invariant torus of X h with frequency ~Z (y) is foliated by mo = n - m dimensional tori. The problem is if there are some lower dimensional tori in the resonant torus survive perturbations. We will see that for most of y E O(g, D) in the measure sense, the answer is yes,i.e., there are some lower dimensional tori on the resonant torus which survive general perturbations. Firstly, we set up the problem. For a given g, by group theory [25], there are integer vectors 1'{,···, 1'~_m E such that is generated by 1'1,"', 1'm, 1'L " " 1'~_m and det(Ko) = 1, where Ko = (K, K'),K = (1'1, "', 1'm), K' = (1'{, ··· , 1'~_m) are n X n, n X m and n X (n - m) respectively. We say h is g-nondegenerate if his nondegenerate and detKT~(y)K f. 0 for y E O(g, D). To catch the main idea, one could keep in mind the typical case h = E~=l y~, 1'i = (0, .. . ,0,1,0, . . · ,0) where 1 is in the i-th position. Since P(x, y) is a real analytic function defined on some complex neighborhood of Tn X D, It admits Fourier's expansion
zn
zn
!
P(x, y)
=
L: PkeFf(k ,x) . kEZ n
For the subgroup 9 of
zn, let
ho(cp, y)
= L: PkeFf(k,x) = kEg
L: PK'leFf(I,cp)
(2.2)
IEz m
where cP = KT x. Clearly, ho has at least m+l critical points on Tm. Moreover, there are at least 2m critical points if all of them are nondegenerate. Let CPo be a nondegenerated critical point of ho(cp,y),i.e.,~(cpo,Y) = 0, and ~2:~Q (CPo, y) is nonsingular, Treshchev proved the following result
416
Jiangong You
Theorem 8 (Treshchev[25],1991 ) For any Yo E O(g, D), if w' = K; w(Yo) as a m-vector satisfies the Diophantine condition (1 . 7) and no eigenvalue of 2 IIKITh82h(Yo)K' is positive or zero where II = 88 \Q(CPO,yo) , there is an - m Y ~ . dimensional torus on the resonant torus which persists perturbatwns and only undergoes a small deformation. Eliasson[14], Chierchia[7]' Rudnev and Wiggins [24] also obtained similar results for the multiplicity one resonant case. Actually, Treshchev's condition implies that the Hamiltonian equations of motion with Hamiltonian h(y) + cho(y, cp) has a hyperbolic (n - m)-dimensional torus for any y E O(g, D) close to Yo. The main idea to deal with hyperbolic lower dimensional tori is, by a series of symplectic changes of variables, to reduce (2.1), in the neighborhood of the hyperbolic lower dimensional torus of h(y) + cho(Y, cp) , to Graff's form [12]. However, this approach does not work when lower dimensional tori of the averaged system h(y) +cho(y,cp) are not hyperbolic. [Cong-Kiipper-Li-You[9], 1999] proved that, each non-degenerate critical point of ho (cp, y) corresponds a family of n - m dimensional tori born from the resonant tori in a resonant surface O(g, D) , no matter if it is hyperbolic or not. Certainly, in this case, the obtained torus might be elliptic, hyperbolic, mixed type or even with multiple normal frequencies . More precisely, [9] considered the system on a whole resonant surface O(g, D) and prove the persistence result of lower dimensional tori for most resonant tori in a measure sense. Theorem 9 (Cong-Kiipper-Li-You[9], 1999) Suppose that H is analytic, h is g-nondegenerate for a given g, and ho (cp, y) has an analytic family of nondegenerate critical point for all y E O(g,D) . Then there is an co> O(depending on h, g, h o} and a Cantor set A' c O(g, D) such that for 0 < c < co, the system (2.1) admits a smooth family( in Whitney's sense} of (n - m)-dimensional invariant tori IYQ parametrized by Yo E O(g, D) . Moreover, the n-m dimensional measure of A' relative to O(g, D) tends to 1 as c --4 O. Since Theorem 9 does not assume the hyperbolicity of the critical points, the following stronger result is an immediate consequence. Theorem 10 (Cong-Kiipper-Li-You[9], 1999) Under the assumptions of Theorem 1, if all critical points of ho (cp, y) are nondegenerate, there is an co > 0 (depending on H o , g, ho}and a Cantor set A' c O(g, D) such that for 0 < c < co, the system (2.1) admits 2m smooth families of (n - m)-dimensional invariant tori parametrized by Yo EA' . Moreover, the measure of A· relative to O(g, D) tends to 1 as c --4 O.
KAM theory for lower dimensional tori
417
Remark 2 Recall the Poincare's theorem ( see, e.g., [4], page 105) on the resonant torus foliated by periodic solutions, i.e., rankg = n - 1 case. He proved that the perturbed system has at least 2n - 1 periodic solutions if all critical points of h o( cP, y) are nondegenerate. Theorem 2 can be regarded as a generalization of Poincare's theorem for periodic solutions to invariant tori case on a Cantor set. Actually, Theorem 1 provides a positive answer to a conjecture about higher dimensional version of Poincare theorem in [4]. Note that in Theorem 8 and Theorem 9, the non-degeneracy conditon on the critical point of the averaged system is assumed, which is believed not to be essential, i.e., H alway have 1+ 1 family of lower dimensional invariant tori burn from the resonant surface O(g, D). Cheng[5], [6] got a positive answer for the multiplicity one resonant case and proved that there are two n - 1dimensional tori born from each resonant torus under a convexity condition of the unperturbed system, but no restriction on the perturbation. Theorem 11 (Cheng[5],[6], 1996,1999) Suppose that 9 is a i-dimensional subh is convex and g-nondegenerated, Then the Hamiltonian {2.1} has group of two families of (n - I)-dimensional tori, one is hyperbolic, another is elliptic.
zn,
If the Hamiltonian (2.1) satisfies H(-x,y) = H(x,y), You [32] proved the persistence of one family of lower dimensional tori for the higher resonant case without any other restrictions except analyticity and smallness. The general higher order resonant case is still open.
Theorem 12 (You[32], 1998) Suppose that 9 is a m dimensional sub-group of his g-nondegenerated and H(x,y) = H(-x,y), then (2.1) has at least one Cantor family of (n - m)-dimensional invariant tori for any 1 ::; m < n .
zn,
3
Infinite dimensional KAM theory
In the 90's the celebrated KAM theory has been successfully extended to infinite dimensional settings so as to deal with certain classes of partial differential equations carrying a Hamiltonian structure, including, as two typical examples, wave equations of the form Utt - U"''''
+ V(x)u =
f(u),
f(u) = 0(u 2 ),
(3.1)
= 0(u 2 );
(3.2)
and nonlinear Schrodinger equations iUt - U"''''
+ V(x)u = f(u),
f(u)
Jiangong You
418
subject to Dirichlet boundary conditions
u(t, O)
= u(t, 1) = 0,
(3.3)
or periodic boundary conditions
u(t, x) = u(t, x
+ 1),
Ut(t, x) = Ut(t, x
+ 1);
(3.4)
see Wayne [26], Kuksin [15] and Poschel [23] . In such papers, KAM theory for lower dimensional tori [20], [19], [13], has been generalized so as to prove the existence of small-amplitude quasi-periodic solutions for (3.1) and (3.2) subject to Dirichlet conditions (on a finite interval for analytic nonlinearities f). Theorem 13 (Wayne[26], 1990, Kuksin[15], 1993) For typical potentials V(x) , the 1D case of nonlinear wave equations and nonlinear Schrodinger equations subject to Dirichlet boundary conditions possess small amplitude quasi-periodic solutions.
Later. Kuksin and Poschel[16] and Poschel[21] proved the first existence result of quasi-periodic solutions for fixed potentials, V = m , for 1D wave equations and 1D schrodinger equations with Dirichlet boundary conditions respectively. The proof of their results based on a kind of infinite dimensional version of the KAM theorem for lower dimensional tori. Recently, the KAM theorem in [31] for lower dimensional tori with multiple normal frequencies has been generalized by Chierchia and You[8] so as to deal with also the 1D wave equations with periodic boundary conditions. In the following we formulate this KAM theorem. Consider the perturbed Hamiltonians
H = N
+P
= (w(~) , J)
1
+ 2" L (nn(€)zn, Zn) + P(O, J , z , Z, ~),
(3.5)
nEN
where (0,1) E Td X R d, Zn E Rdn , ~ EO, nn Ware dn x dn matrices. Our goal is to prove that, for most values of parameter ~ E 0 (in Lebesgue measure sense), the Hamiltonian H = N + P still admits an invariant torus provided IIXpl1 is sufficiently small. In order to obtain this kind of result we shall need the following assumptions on nn and the perturbation P . (AI) Asymptotics of eigenvalues: There exist dEN, 8> 0 and b ~ 1 such that dn ::; d for all n , and (3.6)
KAM theory for lower dimensional tori
419
where An are real and independent of ~ while En may depend on ~j furthermore, the behavior of An 'S is assumed to be as follows Am - An -0 b b = 1 + o(n ), m -n
(A2) Gap condition: There exists
()l
n < m.
(3.7)
> 0 such that
(0'(-) denotes "spectrum of ''' ). Note that, for large i, j, the gap condition follows from the asymptotic property. (A3) Smooth dependence on parameters: All entries of En are smooth functions of ~ . (A4) Non-resonance condition: meas{~ EO:
(k,w(~))«k,w(~))
+ A)«k,w(~)) + A+ JL) = O} = 0,
(3.8)
for each 0 '" k E Zd and for any A, JL E Un EN a(On) j meas == Lebesgue measure. (AS) Regularity of the perturbation: The perturbation P is regular and small in some weighted norm sense (See P6schel[23j for details) . Roughly speaking, regularity means Xp maps a sequence with decay to a sequence with faster decay. Now we can state the KAM Theorem. Theorem 14 ( Cherchia and You[8], 1999) Assume that N satisfies (A1) (A4) and P is regular and small in the sense of (AS) and let 'Y > O. There exists a positive constant € = €( d, d, b, (), ()l , a- a, L, 'Y) such that if IIXp II < € , then the following holds true. There exists a Cantor set 0"'( C 0 with meas( 0 \ O"'() -+ 0 as 'Y -+ 0, such that the Hamiltonian equations governed by H = N + P on the Cantor set 0"'( has d-dimensional H -invariant torus.
The above Theorem has been obtained by P6schel[23j for d = 1, which applies to some ID PDEs with Dirichlet boundary conditions. The main idea has appeared earlier in, i.g., Kuksin[17]' Wayne[26j . Theorem 14 can be applied to the construction of the quasi-periodic solutions for ID wave equations with periodic boundary conditions.
Jiangong You
420
Theorem 15 (Cherchia and You[8], 1999) Consider a family of lD nonlinear wave equation {3.1} subject to periodic boundary conditions, parameterized by ~ == W E 0 ( ~ is chosen to be a fixed family of the eigenvalues) with V (-, ~) real-analytic {respectively, smooth}. Then for any 0 < 'Y « 1, there is a subset 0"( of 0 with meas(O\O"() -+ 0 as'Y -+ 0, such that (3.1)~Eo'Y has a family of small-amplitude {proportional to some power of 'Y}, analytic {respectively, smooth} quasi-periodic solutions of the form
n
where
Un:
r
d
-+ R and w~, ,·· ,w~ are close to
WI, '"
,Wd.
The existence result for (3.1) with Dirichlet boundary condition, obtained by Wayne and Kuksin, has been well known .
4
Bourgain-Craig-Wayne method
Before You[31] and Chierchia and You[8], people generally believed that the KAM theory was not compatible with multiple normal frequencies and thus could not be applied to ID Hamiltonian PDEs with periodic boundary conditions and higher dimensional case where the multiplicity of normal frequencies is essential. [Craig and Wayne, 1993] introduced a new approach to overcome the difficulties caused the multiplicity. They succeeded in proving the existence of periodic solutions of ID wave equations with the periodic boundary conditions. By improving in an essential way Craig and Wayne's technique, Bourgain proved the exisitence of quasi-periodic solutions for perturbed ID nonlinear Schrodinger [Bourgain, 1994]' ID wave equations and, most notably, 2D Schrodinger equations [Bourgain, 1998]. Their approach is based on a Liaponov.. Schmidt decomposition, which involves a multiscale analysis for controlling a Green's function. In this small survey, we shall not try to give a detailed exposition for this significant new approach since it is quite different from the standard KAM approach. In the following, we only state a recent result obtained by [Bourgain, 1998] to show the power of Bourgain-Craig-Wayne method. For details , we refere to [Craig and Wayne,1993], [Craig, 1996] and [Bourgain, 1998]. Consider the Schrodinger equation
.au at -
I
!':!.u + Muu
au
+ f au'
u
= u(x, t), x E T2,
(4.1)
KAM theory for lower dimensional tori
421
where H = E ajlUjUI is real analytic near 0; MCT is a Fourier multiple with MCTei(n,x) = (Tjei(n,x) if n = nj, j = 1,2" .. ,d, MCTei(n,x) = 0 otherwise. Take (T as parameters which varies over a bounded set 0 with positive measure. Theorem 16 [Bourgain, 1998] Let uo(x, t) = E:=l aje i ((n j ,x)+l' j t) , where J.t = Injl + (Tj and aj E R \ {o} . If E lajland 10 are small enough, then there exists a set 0(10, a) C 0, with limHo measure 0 \ 0(10, a) = 0, such that for any (T E 0(10, a), equation (4.1) has a quasi-periodic solution u(x, t), with frequencies Aj '" J.tj, which is close to Uo .
Final remark. So far, whether Theorem 16 has higher dimensional extention is still open. Also there is no counterpart of the result for 2D nonlinear wave equations. The reason has been explained in [Bourgain, 1998]. We hope that the combination of the KAM method and the Bourgain-Craig-Wayne method would be helpful for the problems.
References [1) Arnold, V. I. , Dynamical Systems III, Encyclopaedia of Mathematical Science, SringerVerlag, 1985. [2) J .Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, International Mathematics Research Notices, 1994, 475-497. [3) Bourgain, J ., Quasiperiodic solutions of Hamiltonian perturbations of 2D linear Schriidinger equations, Annals 0/ Mathematics,148(1998) , 363-439. [4) Broer, Hendrik W . Huitema, George B. Sevryuk, Mikhail B. Quasi-periodic motions in families of dynamical systems: order amidst chaos, LNM 1456, 1996. [5) Cheng, C.Q, Birkhoff-Kolmogorov-Arnold-Moser tori in convex Hamiltoniao systems Communication in Math. Phys. 171,1996,529-559. [6) Cheng C.Q., Lower dimensional invariant tori in the region of instability for nearly integrable Hamiltonian systems, Commun. Math .Phys., 203(1999), 385-419. [7) Chierchia L., and Gallavotti G., Drift and diffusion in phase space, Ann. Inst. H. Poincare Phy. Th ., 69(1994), 1-144. [8) Chierchia L. and You J., KAM Tori for 1D Nonlinear wave equations with periodic boundary conditions, To appear in Communication in Mathematical Physics, 2000 . [9) Cong F ., Kiipper T ., Li Y., You J., KAM-Type theorem on resonant surfaces for nearly integrable Hamiltoniao systems, To appear in J. Nonlinear Science, 1999. [10) Craig, W., KAM theory in infinite dimensions, Lectures in Applied Mathematics, 31, 1996. [11) Craig, W., and Wayne, C.K, Newton's method and periodic solutions of nonlinear wave equations, Commun. Pure Appl. Math ., 46, 1993, 1409 -1498.
422
Jiangong You
[12] Graff, S.M., On the continuation of stable invariant tori for Hamiltonian systems,J. Differential Equations, 15, 1974, 1-69. [13] Eliasson L.H., Perturbations of stable invariant tori for Hamiltonian systems, Ann. Sc. Norm . Sup. Psia 15, 1988, 115-147. [14] Eliasson L .H., Biasymptotic solutions of perturbed integrable Hamiltonian systems, Bol. Soc. Mat. 25(1994), 57-76 [15] Kuksin , S.B., Nearly integrable infinite dimensional Hamiltonian systems, Lecture Notes in Mathematics, Springer, Berlin, 1556, 1993. [16] Kuksin, S.B ., Poschel, J., Invariant cantor manifolds of quasiperiodic oscillations for a nonlinear Schrodinger equation, Annals of Mathematics, 142,1995149-179. [17] Kuksin S.B., Perturbation theory for quasi-periodic solutions of infinite-dimensional Hamiltonian systems, and its applications to the Korteweg-de Vries equation, Matern. Sbornik 136(118):31988, English transl. in Math. USSR Sbornik 64 (1994) , 397-413. [18] Lancaster P ., Theory of Matrices, Academic Press LTD, New York and London, 1969. [19] Melnikov V.K., On some cases of the conservation of conditionally periodic motions under a small change of the Hamiltonian function, Soviet Mathematics Doklady, 6, 1965, 1592-1596. [20] Moser J., Convergent series expansions for quasiperiodic motions, Math. Ann., 169(1), 1967, 136-176. [21] poschel,J ., Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. Helvetici, 11, 1996, 269- 296. [22] Poschel J., On elliptic lower dimensional tori in Hamiltonian systems, Math . Z., 202, 1989, 559- 608. [23] Poschel J., A KAM-Theorem for some Nonlinear Partial Differential Equations, Ann. Scuola Norm. Sup. Pisa CI. Sci., 23, 1996, 119-148. [24] Rudnev M . and Wiggins S., KAM theory near multiplicity one resonant surfaces in perturbations of A-priori stable Hamiltonian systems, J. Nonlinear Science 1(1997), 177-209. [25] Treshchev D .V., Mechanism for destroying resonance tori of Hamiltonian systems, Mat . USSR. Sb. 180(1989), 1325-1346 [26] Wayne , C.E., Periodic and quasi-periodic solutions for nonlinear wave equations via KAM theory, Comm. Math . Phys., 121, 1990, 479-528 . [27] Xu J ., Persistence of elliptic lower dimensional invariant tori for small perturbation of degenerate integrable Hamiltonian systems, J. Math. Anal. and Appl. , 208(1997), 372-387. [28] Xu J . and You J ., A symplectic map and its application to persistence of lower dimensional invariant tori for nearly integrable Hamiltonian systems, Preprint, 1999. [29] Xu J. and You J., Persistence of lower dimensional tori under the first Melnikov's conditions, Preprint, 1999. [30] You J., A KAM theorem for hyperbolic-type degenerate lower dimensional tori in Hamiltonian systems, Commun. Math. Phys., 192( 1998), 145-168. [31] You J., Perturbation of lower dimensional tori for Hamiltonian systems, J. Differential Equations,152( 1999), 1-29. [32] You J ., Lower dimensional tori of reversible Hamiltonian systems in the resonant zone, To appear in Proceedings of the Conference in dynamical systems in Memory of Prof. Liao Shantao, World Scientific, 1998.
Periodic Solutions for N-Body Problems* Shiqing Zhangt Department of Applied Mathematics, Chongqing University Chongqing 400044, People's Republic of China Dedicated to the 60th birthday of Professor Paul Rabinowitz
Abstract For Newtonian N-body problems with any positive masses we prove the isolated properties of the collisions for the generalized solution corresponding to the critical point of the Lagrangian action, and we also prove that the minimizer of the Lagrangian action defined on the periodic orbit space on which each body has equal integral mean has no binary and triple collisions neither other collisions under some assumptions.
1
Introduction and Main Results.
N-body problems are related to solving the Newtonian equations:
.. miqi
aU(q)
= -a--' qi
i = 1, ... ,N,
(1.1)
where mi (i = 1, ... , N) is the mass of the i-th body and qi E Rk is the position vector of the i-th body, , q = (ql, .. . , qN), U(q) is the Newtonian potential (1.2) References [2], [4]-[8], [10], [13], [14], [18], [21] used the variational methods to study the solutions of (1.1)-(1.2) . In particular, for the case k = 3, Dell'Antonio ([8]) proved that the minimizer of the Lagrangian action defined on anti-T /2 periodic functional space has no isolated simultaneous collision under some assumptions on the masses of N -bodies, and pointed out some ideas for proving that the minimizer has no isolated two- or three-body collision. • Partially supported by the NSF of China. tThe author would like to thank Professors Long Yiming and Paul Rabinowitz .
423
Shiqing Zhang
424
In this paper, we firstly prove that all kinds of collisions for the critical point of the Lagrangian action are weakly isolated and then generalize the result of Dell'Antonio ([8]). Definition 1.1 ([4], [2]) Given r > 0, let qi E W 1 ,2([0, r], Rk), we call q = (ql, ... , qN) is a generalized solution of (1.1 )-(1.2), if there hold: (i) S(q) = {t E [0, r] I :11 :::; i i= j :::; N, s.t., qi(t) = qj(t)} has zero Lebesgue measure. (ii) For all t E [0, r] \ S(q), q(t) satisfies (1.1). (iii) For all t E [0, r] \ S(q), there holds
Definition 1.2 A collision time to E [0, r] is called weakly isolated or isolated if there is a neighbourhood of to such that there is no the same kind of collisions or no collisions in the neighbourhood of to. Theorem 1.1 The collisions of the generalized solution for (1.1)-(1.2) are weakly isolated. Given T > 0, let
(1.3)
1
r L mdcii(tWdt + inr U(q)dt T
f(q) = 2" in
o
T
N
i=l
(1.4)
0
Definition 1.3 ([8]) we call (1.1)-(1.2) satisfy the condition (Hjd: For the givep masses ml,· .. , mK, every nonplanar central configuration ([1], [3], [15]. [19], [20]) for K bodies is the isolated modulo rotation. Theorem 1.2 The minimizer q(t) of f(q) on A has no binary and triple collisions; and if condition (H K) holds, then the minimizer has no collision for K-bodies.
2
The Proof of Theorem 1.1.
Lemma 2.1 The binary collisions of the generalized solution for Newtonian N -body problems are weakly isolated.
Periodic solutions for N-body problems
425
Proof Without loss of generality, we assume t = 0 is a binary collision time for m1 and m2. We claim that t = 0 is a weakly isolated collision time. Otherwise, there is a sequence tn -+ 0 such that d12(t) = Iq1(t) - q2(t)1 satisfies d12 (t n ) = O. Hence d12 (t) attains the maximum value at some in E (tn, tn+d, so we have 2 (2.1) d d12 (t) t=tn ::; 0 dt2
I
Let ~ = ~(t) =
(2.2)
q1(t) - q2(t)
Then (1~12)' = 2~ . ~
:2 (1~12) =
(2.3)
21~12 + 2~ . ~
(2.4)
By the first two equations of the system (1.1) we know that (2.5)
Hence
~(1~12) = 21~12 _ 2(m1 + m2) dt 2 Iq1 - q21 In the following, we will further prove that ~1~12 _ m1 + m2 2
d 12
+ 0(1),
= 0(1)
(2.6)
(2.7)
We use the energy formula N
h =
~L
m;!qil 2
-
U(q)
i=1
(2.8)
Shiqing Zhang
426
Summing the first two equations of the system (1.1), we have
(2.9) By the last N - 2 equations of (1.1), we have
miqi(t)
E
C 2, as t
-t
0
(2.10)
By (2.8)-(2.10) we have (2.7) . Hence by (2.6) and (2.7) we have
~( leI 2) = 2(ml + m2) + 0(1)
dt 2
d 12
(2 .11)
Then (2.11) contradicts with (2.1) since from (2.11) we have
d2
dt 2 (leI 2 )
-t +00
as t
-t
0
(2.12)
Lemma 2.2 The triple collisions of the generalized solution of (1.1)-(1.2) are weakly isolated. Proof Without loss of generality, we assume t = 0 is a triple collision time for ml, m2 and m3. If t = 0 is not weakly isolated, then there is a sequence tn -t 0 of the triple collision times such that
423(t)
=
L
m,mjlq,(t) - qj(t)12
(2.13)
1~i<j9
=
=
satisfies d'f23(tn) d'f23(t n+1 ) 0, hence d'f23(t) attains the maximum value at some tn E (tn, tn+d, so we have (2.14) For 1
~
i
<j
~
3, let (2.15)
Then
We notice that
Periodic solutions for N-body problems
427
Where (2.17)
We notice that
Shiqing Zhang
428
(2.20) We use the energy representation: h
=
= (2.21) Summing the first three equations of the system (1.1), we know that 3
Lmiqi(t) i=l
E
C 2, as t -+ 0
(2.22)
By the i-th (4 ::; i ::; N) equation, we know that
miqi(t) E C 2, as t -+ 0
(2.23)
By (2.20)-(2.23) we have
~
m L m ·m 'I{ .12 + M3 (-m 1 m 2 - mlm3 - m2 3 ) - 0(1) (2.24) 2
Hence from (2.24) and (2.19) we have 1
d?-
2
2' dt2 (d123 (t)) -+
M3 (m 1 m 2 Iql - q21 +00 as t -+ 0
+
mlm3 Iql - q31
+
m2 m 3 ) Iq2 - q31
+ 0(1) (2.25)
Obviously, (2.25) is contradiction with (2.14) , so the proof is completed. Lemma 2.3 If 4 ::; K ::; N - 1, then the collisions of K -bodies corresponding to the generalized solution of (1.1) are weakly isolated.
Periodic solutions for N-body problems
429
Proof Without loss of generality, we can assume t = 0 is a collision time for ml, m2, . .. , m K . If t = 0 is not weakly isolated, then there is a sequence tn --t 0 of the collision with the K -bodies such that A(t) =
L
(2 .26)
mimjlqi(t) - qj(tW
1:9<j~K
satisfies (2.27)
Hence A(t) attains the maximum value at some in E (tn, t n+1), so we have (2.28)
For 1
~
i
<j
~
K, let
(2.29)
Then
1 cP 2 2mimj dt 2 (I~ij I )
We observe the sum:
.
2
..
= mimj I~ij I + mimj~ij . ~ij
L
mimj~ij' tij
(2 .30) (2.31)
l~i<j~K
The term mlm2~12 . t12 includes (2.32)
(2.33)
(2.34)
(2.35)
Shiqing Zhang
430
(2.36) Summing (2.35) and (2.36) we have -1 m 1m 2m j
lql
- q2
I
(2.37)
More generally, for 1 ::; l < h ::; K , m/mh6h . ~/h includes mlmh
-(m/
+ mh)
(2.38)
Iql - qhl
(2.39)
(2.40) Summing (2.39) and (2.40) we have -1 m1mhm j
l q/ - qh
I
(2.41)
From the above analysis, we know that the sum (2 .31) includes
(2.42)
Periodic solutions for N-body problems
431
Where (2.43) i=l
We notice that except (2.42) the terms in (2.31) are bounded. We notice that
1"
.
~ mimj!~ij! _ 3_ 2 l
2
~ ~MK t, m,l.,I' - ~ It, m,.",
(2.44)
We use the energy representation: h
= = (2.45)
Summing the first K equations of the system (1.1), we have K
Lmiqi(t) E C i=l
By the i-th (K
+1~ i
~
2
,
as t -+ 0
(2.46)
N) equation, we have miqi(t) E C
2
,
(2.47)
as t -+ 0
By (2.44)-(2.47) we have
1" -2 ~
.
< 15,i
mimj!~ij!
2
+ MK
, , - m i m j -_ 0(1)
~
(2.48)
l
Hence from (2.42), (2.48) and (2.30) we have
1" 2' ~
rP!
mimj dt 2 (~ij! 2 )
15,i<j5,K
=
j MK ( L mim ) 15,i
-+ +00 as t -+ 0
+ 0(1) (2.49)
Shiqing Zhang
432
The proof is completed since (2.49) contradicts with (2.28). Lemma 2.4 The simultaneous collosions for the generalized solution of (1.1) are weakly isolated. Proof Without loss of generality, we assume t = 0 is a simultaneous collision time. We claim that t = 0 is a weakly isolated collision. Otherwise, there is a sequence tn -t 0 of simultaneous collision times such that (2.50) B(t) = mimjlqi(t) - qj(t)1 2
L
l~ i "#j~N
satisfies
B(t n ) = B(t n +1) = 0
(2.51)
Then B(t) attains the maximum value, at some in E (tn, t n+1), so (2.52) By Lagrange's identity, we have (2.53)
Where
N
M
N
= Lmi'
1=
i=1
Lmilqil2
(2.54)
i=1
We notice that
! Itmiqil2 =
2 (tmiqi) (tmiqi)
1=1
:2 It t-l
miqil2 =
21t
miqil2
1=1
2
+ ( t m iqi ) 1=1
t_l
(2.55)
.=1
(t
m iiii)
(2.56)
1=1
Summing all equations of the system (1.1), we have N
L miQi(t) == 0
(2.57)
i=1
Hence
N
L miqi(t) E C 2 i=1
(2.58)
Periodic solutions for N-body problems
433
By (2.56)-(2 .58) , we know
(2.59) On the other hand, by Lagrange's formula ([3), [19]) we have j = 4h + 2U
Where h is the energy of (1.1)-(1.2) . Hence when t j
(2.60) ~
0, we have
= 4h + 2U ~ +00
(2.61)
From (2.61) , (2.59) and (2.53) we have B(t) ~ +00, as t ~ 0
(2 .62)
The Lemma 2.4 is proved by (2.62) and (2.52) . Lemma 2.5 Assume q(t) = (ql (t), ... , qN(t)) is a generalized solution for the system (1.1), when t = O,ml , ... ,mK collide together, mKH, .. . ,ml collide together and mlH, . . . ,mN are free-collision. Then t = 0 is a weakly isolated collision time for ml , ... , m K and m K H , ... , mi. Proof If t = 0 is not a weakly isolated collision time, then there are sequences tn ~ 0, Sn ~ 0 such that
G(t) =
L
mim j lqi(t) - qj(t)12
(2.63)
l~ i <j~K
D(t) =
(2.64)
satisfies
G(tn) = G(tnH) = 0
(2.65)
D(sn) = D(sn+d = 0
(2.66)
Hence G(t) and D(t) attains the maximum values respectively at tn E (tn, tnH) and sn E (sn' SnH). Then (2.67) Let (2.68)
Shiqing Zhang
434
Then (2.69)
L
mimj~ij
. tij
=
(2.70)
MKI
K+1~i<j9
Where
K
MK
= L mi, i=l
I
MKI
=
L
mi
(2.71)
i=K+l
Hence
(2.72) We notice that
(2.73)
I
1
2
K+1~i<j9
L (2.74)
We use the energy representation: h
=
Periodic solutions "for N-body problems
=
435
~K [~MK t.=1 mil4il 2- ~ Itmi4il2] .=1 +
~KI [~MKI. t
2- ~ I mi4il2] mil4il .=K+1 i=K+1
t
+2~K Itmi4il2 + 2~ .=1 .
I
t
KI i=K+1
N
L miltiil 2+ L
+~
1~i<j~K
2 i=l+l
+
=
mitiil2
L
K+1~i<j9
-mimj Iqi - qjl
-mimj Iqi - qjl
+ 0(1)
(2.75)
~ (~ L mimjl~ijI2) + L 1~i<j~K
K
1~i<j~K
-mimj Iqi - qjl
+
~KI (~ K+1~i<j~1 L mimjl~ijI2) + L -mimj K+l~i<j9 Iqi - qjl
+
2~
It mitiil2 + 2~KI Ii=K+1 mi4il2
t
K i=1
1
+2"
N
L miltii!2 + 0(1)
(2.76)
i=I+1
Summing the first K equations of the system (1.1), we have that K
K
L miiii(t) E C, L miqi(t) E C i=1
2
,
as t -t 0
(2.77)
i=l
Summing the equations from number (K + 1) to number (I) in the system (1.1), we have I
L
miqi(t) E
C 2, as t -t 0
(2.78)
i=K+l
From the i-th (I
+ 1 :::; i
:::; N) equation in the system (Ll), we have (2.79)
Shiqing Zhang
436
From (2.76)-(2.79) we have
(2.80)
From (2.80) and (2.72) we have 1 1··
1 1
..
2 MK C(t) + 2 MKI D(t)
L
mimj.
l~i<j~K Iq, - qJI
+
L
m~mj. qJI
(2.81)
K+l
Hence we have 1··
1··
--C(t) + --D(t) -+ +00 as t -+ 0 2MK 2MKI
(2.82)
(2.82) is contradication with (2.67), so Lemma 2.5 is proved. Similar to the proof of Lemma 2.5, we have the more general result, that is, the Theorem 1.1.
3
The Proof of Theorem 1.2.
Lemma 3.1 The functional f(q) in (1.4) attains its global minimum value on the colsure X of A, and q(t) = (ql (t), .. . , qN(t)) the minimizer is a generalized T-periodic solution for systems (1.1)-(1.2) . Proof We notice that the functional f(q) is invariant under the translation ql (t)dt = ... = qN(t)dt, so we can work f(q) on
J:
J:
Ao =
{q = (ql, ... ,qN) E AlloT qi(t)dt = O,i = 1, ... ,N }
(3.1)
It's easy to prove that f(q) is coercive and weakly lower-semi-continuous on f(q) attains the minimum value on and X, furthermore, similar to the proof of (6) and [10], we know that the minimum point q(t) is a generalized solution for systems (1.1)-(1.2). By Theorem1.1 and the globally minimizing property for q(t),we have Lemma 3.2 The collisions for the minimizer q(t) are isolated. Lemma 3.3 The minimizer q(t) has no binary and triple collisions.
Ao, so
Ao
Periodic solutions for N-body problems
437
Proof Assume q(t) has binary or triple collisions. Then by Lemma 3.2, the collisions are isolated, this allows us to construct a perturbation q€(t) of q(t) such that f(q€(t)) < f(q(t)) (3.2)
Without loss of generality, we can assume t time for m1 and m2 or m1,m2 and m3 . Set el(t)
= 0 is a binary or triple collision
= Itl- 2 / 3 ql(t), l = 1, . .. ,n(n = 2 or 3)
(3.3)
Then there exist limits ( [19],[20]) : lim (ei(t) - el(t))
t--+O+
Let 17
= ej", t--+olim (ei(t) -
el(t))
= Cit
(3.4)
= 17(t) = (171 (t), . .. , 17N(t)), 17n+l (t) = ... = 17N(t) = 0, where 17i(t) = w(t)(i,i = 1,···,n
(3.5)
w(t) E W 1 ,2(R/T Z, Rk) satisfies
w(t)
=
1,
o ::; t ::; co,
cp(t), 0, -cp(T - t),
co ::; t ::; 2co, 2co ::; t ::; T - 2co T - 2co ::; t ::; T - co T - co ::; t ::; T
1
-1,
sup Iw(t)1 09~T
(3.6)
= 1,
(3.7)
(i is chosen such that
I(i - (II
> 0, 1::; i f
lim ((i - (I,ei(t) - 6(t))
t--+O±
(3.8)
l ::; n
= ((i -
(I,e~) :::: 0
(3.9)
We claim (i satisfying (3.8) and (3.9) does exist. In fact, for n = 2, we can choose (1 and (2 such that the angles between (1 - (2 f 0 and and 12 are less than 90 0 , this can be done if we choose (1 - (2 to lie on the straight line which splits the angle between and e12 into two equal angles. For n = 3, we can choose (1, (2 and (3 such that (1 - (I (l = 2, 3) to lie on the straight line which splits the angle between e~ and into two equal angles, then we have (3.10)
ei;
e
ei;
eli
Shiqing Zhang
438
Furthermore, we can choose (1 - (2 and (1 - (3 such that 1(1 - (21 =F 1(1 - (31, that is, (3.8) holds. From (3.10) we have
(3.11) Now we estimate
f(q(t) Where
h
=~
+ crJ(t)) - f(q(t)) = h + h
1L T
o
(3.12)
N
mi(ltii
+ c1jd 2 -ltiiI 2 )dt
(3.13)
i=l
12 = loT (U(q
+ crJ) - U(q))dt
(3.14)
First we notice that
By the properties of w(t), we have that
If =
~ (1:°0 + l~~::)
t
mi[2cw(t) (tii, (i)
+c
2 2 2 W (t)l(iI jdt
< (Clc· cOl + C2c 2 . co 2)co = Clc + C2c2cOl
(3.16)
Now we estimate
=-1
1
o
(
L 19<j::;n
1Tmimj(qi-qj+SCrJi-SCrJj'CrJi-CrJj)dt)dS 0 Iqi - qj + ScrJi - scrJjl3
(3.17)
By the properties (3.6), (3.8) and (3.9) of W, (i and ~i(i = 1, ... , n) we have
Let (3.19)
Periodic solutions for N-body problems
439
Then
(3.20)
Where w=w(c,r)=w
(3.21)
Hence by (3.7) we have sup{ w(c, r), 0 ~ r ~ 2coc-3/2}
=1
(3.22)
It is well known ([8], [19], [20]) that for the isolated binary or triple collision, there is constant C > 0 such that for 1 ~ i "I l ~ n,
(3.23)
We set I(i - (d
= Cil > 0
(3.24)
Then for 0 ~ r ~ 2coc-3/2, we have
r 4/ 3 2C2
sC~
+ -2-
< Ir2/3(~i - ~l) + SW((i - (IW < 2C2r 4 / 3 + 2sclL
(3.25)
Then
Notice that for 2coc- 3/ 2 -+ +00, the integral: 1
1
ds
12000-3/2
C2 S il dr 12C2r 4/3 2sClL 13 / 2
+
(3.27)
converges absolutely to a positive constant. Hence there is C3 > 0 such that (3.28)
Shiqing Zhang
440 We notice that for 2eOe- 3 / 2 ~
+00,
(3.29) We notice that
(3.30) Where (3.31)
= ( (2e
io
o
+
(T) iT-2eo
L
[mim j _ mimj ] dt l
(3.32)
I
Since at t = 0 and t = T, ql, ... , qn collide together, but qn+1,···, qN are collision-free and don't take part in the collision with ql, ... ,qn, hence for 1 :::; i :::; n, n + 1 :::; j :::; N we have
mimj
Iqi
+ e1Ji - qjl
_ mimj Iqi - qjl
= 0(1)
as t ~ O,e ~ O.
(3.33)
Hence from (3.33) and (3.32) we have
(3.34) From (3.12), (3.15), (3.16), (3.28) and (3.29) we have
f(q
+ e1J) - f(q) :::; O(e) - C3 ../€ + O(eo)
(3.35)
We can choose co such that (3.36) then
f(q
+ e1J) - f(q) :::; O(e) -
C 3 ../€ + 0(e 2 / 3 )
< O,e ~
(3.37)
0
This is a contradiction, hence q has neither binary nor triple collision. Lemma 3.4 The minimizer q(t) has no K-bodies collision if the condition (Hi<) holds. Proof If the Lemma 3.4 is not true, then without loss of generality , we can assume ml, ... , mK collide together at t = O. By Lemma 3.2, t = 0 is isolated, hence by the assumption (Hi<) (Dell' Antonio [8]) we know lim (~i(t) - ~j(t))
t-+o*
2 = t-+O* lim Itl- / 3 (qi(t) -
qj(t))
= ~;3·' 1:::; i,j :::;
K
(3.38)
Periodic solutions for N-body problems
do exist and there is C
441
> 1 and C1 small enough such that for 1 ~ i ::f. j
~
K,
(3.39) Similar to the proof of Lemma 3.3, we construct a perturbation qe(t) of q(t) such that f(qe(t)) < f(q(t)) (3.40) Let 77 = 77(t) = (771 (t), ... , 77K(t), 0, ... ,0)
(3.41)
77i(t) = w(tK, i = 1, . . . ,K
(3.42)
where Where w(t) is the same as (3.6) and (3.7) of Lemma 3.3. (i(i chosen such that I(i - (jl > 0, 1 ~ i ::f. j ~ K ((i - (j,~m
:2: 0, 1 ~ i::f. j ~ K
= 1, . . . ,K)
is
(3.43) (3.44)
The (1, . . . ,(K satisfying (3.43) and (3.44) do exist. In fact, we can choose (1 - (j(2 ~ j ~ K) to lie on the straight line which splits the angle between and ~lj into two equal angles. Furthermore, we can choose (1 - (j such that
f0
(3.45) Then (3.46) and (3.47) From (3.45) we also have ((i - (1'~~)
0
(3.48)
:2: 0
(3.49)
:2:
From (3.45) and (3.46) we have ((i - (j,~m
Since the total number for vectors (1 - (2, ... , (1 - (K is K - 1, so we can choose (1 freely, hence {(1, ... ,(K} satisfying (3.43) and (3.44) do exist. Similar to the proof of Lemma 3.3 we can find a qe(t) such that (3.40) holds, this is impossible.
442
Shiqing Zhang
References [1] R. Abraham and J. Marsden, Foundations of Mechanics, 2nd ed. Benjamin/Cummings, London, 1978. [2] A. Ambrosetti and V. Coti Zelati, Periodic Solutions of Singular Lagrangian Systems. Birkhauser, Basel. (1993). [3] V. Arnold, V. Kozlov, and A. Neishtadt, Dynamical Systems, III. Mathematical Aspects of Classical and Celestial Mechanics. Russian ed. (1985) , Spinger. Berlin. English ed. (1988) . [4] A. Bahri and P. Rabinowitz, Periodic solutions of Hamiltonian systems of thre-body type. Ann. IHP. Anal. non lineaire. 8. (1991) 561-649. [5] A. Chenciner and N. Desolneux, Minima de l'integrale d'action et equilibres relatifs de n corps. C. R. Acad. Sci. Paris, Sr. 1. Math. 326. (1998) 1209-1212. [6] V. Coti Zelati, The periodic solutions of N-body type problems. Ann. IHP. Anal. non lineaire. 7. (1990) 477-492. [7] M. Degiovanni and F. Giannoni, Dynamical systems with Newtonian type potentials. Ann. Scuola Norm. Super. Pisa. 15. (1989) 467-494. [8] G. F. Dell'Antonio, Classical solutions of a perturbed N-body system. In Topological Nonlinear Analysis II. M. Matzeu and A. Vignoli ed. Birkhauser. (1997) 1-86. [9] L. Euler, De motu rectilineo trium corp6rum se mutuo attrahentium. Novi. Comm. Acad. Sci. Imp. Petropll. (1767) 145-151. [10] W. Gordon, A minimizing property of Keplerian orbits. Amer. J. Math. 99. (1977) 961-971. [11] G. Hardy, J . Littlewood, and G. Polya, Inequalities. 2nd ed. University Press, Cambridge (1952). [12] J. Lagrange, Essai sur Ie probleme des trois corps. (1772) . Ouvres, Vol. 3 (1783). 229-331. [13] Y. Long and S. Zhang, Geometric characterizations for variational minimization solutions of the 3-body problem. Nankai Inst. of Math. Preprint. No. 1998-M-005, June (1998). Acta Math. Sinica. to appear.
Periodic solutions for N-body problems
443
[14] Y. Long and S. Zhang, Geometric characterizations for variational minimization solutions of the 3-body problem with fixed energy. J. Diff. Equa. to appear. [15] K. Meyer and G. Hall, Introduction to Hamiltonian Systems and the Nbody problems. Springer. Berlin. (1992). [16] H.Pollard,Celestial mechanics,AMS,1976. [17] P. Rabinowitz, Periodic solutions of Hamiltonian systems. Comm. Pure Appl. Math. 31. (1978). 157-184. [18] E. Serra and S. Terracini, Collisionless periodic solutions to some threebody problems. Arch. Rational Mech. Anal. 120 (1992), 305-325. [19] C. Siegel and J. Moser, Lectures on Celestial Mechanics. Springer. Berlin. (1971). [20] A. Wintner, Analytical Foundations of Celestial Mechanics. Princeton University Press. Princeton. (1941). [21] S. Zhang, Variational minimizing properties for N-body problems, Proc. in homor of Liao Santao, World Science Pub., to appear. First manuscript June 28, 1999
List of Scientific and Organizing Committees Scientific Committee: Kung-Ching Chang (Chairman) Paul Rabinowitz Zhihong Xia Organizing Committee: Yiming Long (Chairman) Chong-Qing Cheng Chun-gen Liu Honghai Lu
List of speakers BAHRI Abbas Permanent Institute:Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA . E-Mail address: [email protected] BARTSCH Thomas Permanent Institute:Mathematisches Institut, Justus-Liebig-Universitaet Giessen, Arndtstrasse 235392 Giessen, Germany. E-Mail address:[email protected] BOLOTIN Sergey Permanent Institute:Department of Mathematics and Mechanics, Moscow State University, Vorobevy Gori(Hills), 119899 Moscow, Russia. E-Mail address:[email protected] CAO Dao-Min Permanent Institute:Institute of Applied Mathematics, Academy of Sciences, Beijing 100080, People's Republic of China.
445
446 E-Mail address:[email protected] CHANG Kung-Ching Permanent Institute:Department of Mathematics, Peking University, Beijing 100871, People's Republic of China. E-Mail address:[email protected] CHENG Chong-Qing Permanent Institute:Department of Mathematics, Nanjing University, Nanjing 210008, People's Republic of China. E-Mail address: [email protected] CHEN Chao-Nien Permanent Institute:Department of Mathematics, National Changhua University of Education, Changhua, Taiwan, ROC. E-Mail address:[email protected] DENG Yinbin Permanent Institute: Department of Mathematics, Huazhong Normal University, Wuhan, 430000, People's Republic of China. E-Mail address:[email protected] DING Yan-Heng Permanent Institute:Institute of Mathematics, Academy of Sciences, Beijing 100080, People's Republic of China. E-Mail address: [email protected] FELMER Patricio Luis Permanent Institute:Departamento de Matematicas, Universidad de C; Casilla 170 Correo 3, Santiago, Chile. E-Mail address:[email protected] IZYDOREK Marek Permanent Institute:Department of Technical Physics and Applied Mathematics, Technical University of Gdansk, 80-952 Gdansk ul.G.Narutowicza 11/12, POLAND. E-Mail address:[email protected] JIAN Huai-Yu Permanent Institute:Department of Applied Mathematics, Tsinghua University, Beijing 100084, People's Republic of China. E-Mail address:[email protected] JIANG Mei-Yue Permanent Institute:Department of Mathematics, Peking University,
447 Beijing 100871, People's Republic of China. E-Mail address:[email protected] LI Gong-Bao Permanent Institute:Wuhan Institute of Physics and Mathematics, Academy of Sciences, Wuhan 430071 , Hubei, P. R. China. E-Mail address:[email protected] LI Jia-yu Permanent Institute:Institute of Mathematics, Academy of Sciences, Beijing 100080, People's Republic of China. E-Mail address:[email protected] LI Shu-Jie Permanent Institute:Institute of Mathematics, Academy of Sciences, Beijing 100080, People's Republic of China. E-Mail address:[email protected] LI Yanyan Permanent Institute:Department of Mathematics, Rutgers University, New Brunswick, NJ 08903, USA. E-Mail address:[email protected] LID Bin Permanent Institute:Department of Mathematics, Peking University, Beijing 100871, People's Republic of China. E-Mail address: [email protected] LID Chun-Gen Permanent Institute:Department of Mathematics, Nankai University, Tianjin 300071, People's Republic of China. E-Mail address:[email protected]@nankai.edu.cn LID Jia-Quan Permanent Institute:Department of Mathematics, Peking University, Beijing 100871, People's Republic of China. E-Mail address:@sxxO.math.pku.edu.cn LONG Yiming Permanent Institute:Nankai Institute of Mathematics, Nankai University, Tianjin 300071, People's Republic of China. E-Mail address:[email protected] LU Guangcun Permanent Institute:Nankai Institute of Mathematics, Nankai University,
448
Tianjin 300071, People's Republic of China. E-Mail address:[email protected] MA Renyi Permanent Institute:Department of Applied Mathematics, Tsinghua University, Beijing 100084, People's Republic of China. E-Mail address:[email protected] MANCINI Giovanni Permanent Institute:Dipartimento di Matematica, Universita di Roma Tre, Largo S. Leonardo Murialdo, 00146 Roma, Italy. E-Mail address: [email protected] MAWHIN Jean L. Permanent Institute: Mathematiques, Universite Catholique de Louvain, chemin du cyclotron, 2, B-1348 Louvain-Ia-Neuve, Belgium. E-Mail address:[email protected] NI Wei-Ming Permanent Institute:School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St.S.E., Minneapolis, MN 55455, United States of America. E-Mail address: [email protected] OUYANG Tiancheng Permanent Institute:Department of Mathematics, Brigham Young University, Provo, Utah, USA. E-Mail address: [email protected] PAN Xingbin Permanent Institute:Department of Mathematics, Zhejiang University, Hangzhou 310027, Zhejiang, People's Republic of China. E-Mail address:[email protected] RABINOWITZ Paul H. Permanent Institute:Department of Mathematics, University of Wisconsin, Van Vleck Hall 709, 480 Lincoln Drive, Madison, Wisconsin 53706, USA. E-Mail address: [email protected] RYBICKI Slawomir Maciej Permanent Institute:Department of Mathematics and Informatics, Nicholas Copernicus University, PL-87-100 Torun, UL.Chopina 12/18, POLAND. E-Mail address:[email protected] SILVA Elves Alves De Barrose
449 Permanent Institute:Universidade De Brasilia, Departamento. De Matematica-IE, Campus Univ. 70910-900, Brasilia, DF, BRAZIL. E-Mail address:[email protected] TANAKA Kazunaga Permanent Institute:Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Okubo Shinjuku-Ku, 169 Tokyo, Japan. E-Mail address:[email protected] WANG Zhi-Qiang Permanent Institute:Department of Mathematics, Utah State University, Logan, UT84322-3900, USA. E-Mail address: [email protected] WU Shao-Ping Permanent Institute:Department of Mathematics, Zhejiang University, Hangzhou 310027, Zhejiang, People's Republic of China. E-Mail address:[email protected]. XIA Zhihong Permanent Institute:Department of Mathematics, Northwestern University, Evenston Ill, USA. E-Mail address: [email protected] YANG Xuefeng Permanent Institute:Department of Mathematics and Statistics, McMaster University, Hamilton, On LBS4K1, Canada. E-Mail address:[email protected] YOU Jian-Gong Permanent Institute:Department of Mathematics, Nanjing University, Nanjing 21000B, People's Republic of China. E-Mail address:[email protected] ZENG Chongchun Permanent Institute: Courant Inst.Math. New York Univ. New York, USA. E-Mail address:[email protected] ZHANG Li-Qun Permanent Institute:lnstitute of Mathematics, Academy of Sciences, Beijing 1000BO, People's Republic of China. E-Mail address:[email protected] ZHANG Shiqing
450 Permanent Institute:Department of Applied Mathematics, Chongqing University, Chongqing 400044, Sichuan, People's Republic of China. E-Mail address: [email protected] ZHOU Feng Permanent Institute: Department of Mathematics, Huadong Normal University, Shanghai, 200062, People's Republic of China. ZHOU Jilin Permanent Institute:Department of Astronomy, Nanjing University, Nanjing 210008, People's Republic of China. E-Mail address:[email protected] ZHU Xi-Ping Permanent Institute:Department of Mathematics, Zhongshan University, Guangzhou 510275, Guangdong, People's Republic of China. E-Mail address: [email protected]