Morse Theoretic Aspects of p-Laplacian Type Operators (Mathematical Surveys and Monographs)

vanishes on Uq+2, so Co(X) = {Co(X),6q} is a subcomplex of C(X). Let

2. BACKGROUND MATERIAL

32

C(X) = C(X)/C0(X) = {Cq(X)/CC(X),bq}. A map f : X y.Y of spaces induces a cochain map f# : C(Y) /- C(X) by (f# gV))(xo,...,

4'E C4(Y) vanishes on If zO e Q0 '(Y) and V is an open covering of Y such that Vq+I, then U = f -I V is an open covering of X and f #q vanishes on xq)=4'(A(x0),..., f(xq)),

So f# maps Co(Y) into Co(X) and hence induces a cochain map f# : C(Y) -* C(X). If i : A c X, then C(X, A) = ker i# ker 6g } is a subcomplex of C(X). Let C(X, A) be the subcomplex of C(X) consisting of functions that are locally zero on A. Then we see that Co(X) c C(X,A) and C(X,A) _ Uq+I.

C(X, A)/Co(X). Define the Alexander-Spanier cohomology of (X, A) by H* (X, A) = H* (C(X, A)).

We write H* (X, Q) = H*(X). If f : (X, A) -. (Y, B) is a map of pairs, then f # : C(Y, B) -. C(X, A) is a cochain map and hence induces linear maps

f* : H*(Y,B)

H*(X,A)

such that (g f )* = f *g* for (X, A) (Y, B) ' (Z, C) and id (x,A) _ idH*(x,A) We refer to Spanier [126] for the proofs of the axioms for this cohomology theory. (cl) Homotopy axiom: If fo ^ f1 : (X, A) -. (Y, B), then

fo =fi :H*(Y,B)-'H*(X:A). (c2) Exactness axiom: Each triple (X, A, B) has an exact sequence Hq(X, A)

3*

Hq(X, B)

Hq(A, B)

Hq+1(X A) where i : (A, B) c (X, B), j : (X, B) c (X, A), and b is called the connecting map. 0 (c3) Excision axiom: If U is an open subset of X such that U c A, then j : (X\U, A\U) c (X, A) induces an isomorphism

j' : H*(X,A)

H*(X\U,A\U).

(c4) Dimension axiom: If X is a one-point space, then (2.2) Hq(X) x SgOZ2. Cohomology is a homotopy invariant, i.e., (X, A)

(Y, B)

H* (X, A)

H* (Y, B).

To see this, let f : (X, A) -. (Y, B) and g : (Y, B)

(X, A) be homotopy

equivalences. By (cl),

f*g* = (gf)* = id(*x,A) = idH*(x,A)

2.3. ALEXANDER-SPANIER COHOMOLOGY THEORY

33

and similarly g* f * = id H*(y B), so g* = (f *)-1. In particular, the cohomology of a contractible space is given by (2.2). If r : X - A is only a retraction and i : A c X, then i*r* = (ri)* = id*A = idH*(A) and hence r* is one-to-one and i* is onto. Example 2.3. Hq(Sn-1) z aqo Z2 O aq,n-1 Z2 When X # 0, it is sometimes more convenient to work with the reduced groups

Hq(X)

H°(X)/Z2, q=0

q%1 Hq(X), which are trivial in all dimensions for contractible spaces. They also fit into an exact sequence Hq-1(A) A) Hq-1(X) ,

- Hq(X

Hq(X) for a pair (X, A) with A # 0. In particular, we have Proposition 2.4. Let (X, A) be a pair with X contractible.

(i) If A = 0, then Hq(X,A)

40 Z2.

(ii) If A # 0, then Hq(X,A) Hq-1(A) In particular, Ho(X, A) = 0.

Example 2.5. Hq(Sa-1)

Vq.

dg n-17 Z2, Hq(D", S -1) , Hq-1(Sn-1)

Sqn Z2

: HP(X) X Hq(X) There is a product HP+q(X), called the cup product, giving H*(X) the structure of a graded ring. To define this product, first define a product : CP(X) X Cq(X) --. CP+q(X) by (cP - 0) (xo, ... , xp+q) = cO(xo, ... , xp) V)(xp, ... , xp+q).

If cp or Vi is locally zero, then so is cp -0, and hence - induces a product -op(X) X Z 71(X) -. C +"(X). It is easy to verify that b(o - l)=8W- 0 +cp- fig/): so if cp and 1/i are cocycles, then so is co - 11i, and if one of them is a coboundary in addition, then so is cp -0. 1(i. Hence -- induces a product on cohomology classes. We write y - W = cp2, etc. Now suppose A1, A2 c X are such that the interiors of Al and A2 in Al u A2 cover the union. Then an element of Cp+q(X) that is locally zero on both Al and A2 is locally zero on Al u A2 also, so we get a cup product : CP(X, Al) X Cq(X, A2) - CP+q(X, Al u A2), which then induces a cohomology product '- : HP(X, A1) X Hq(X, A2) HP+q(X, Al U A2).

2. BACKGROUND MATERIAL

34

Example 2.6. H* (RPG0) = Z2 [w], the polynomial ring on a single generator w e H1(RPCO), and H*(1RP"-I) = Z2[w]/(w"), the quotient of Z2[w] by the ideal generated by w".

A neighborhood of a pair (A, B) in X is a pair (U, V) in X such that U is a neighborhood of A and V is a neighborhood of B. The set A of all neighborhoods of (A, B) is partially ordered downward by inclusion: (Ux,Vx) < (Un,VV) if iaN, : (Uu,Vµ) c (UU,V)). Since (Ua n U1 ,VV, n VN,) c (UA,VV), (U,,, V.), we see that A is directed. The collection {Hq(UT, VA)}AEA of cohomology groups and the induced maps i* : Hq(Ua,VA) -> Hq(UT,V4) is a directed system since iTa = id* UA vA) = id H4(U,,,va) and i*

i** iaµ. The maps j,* : HQ (U,\, VA) Hq(A, B) induced by the inclusions IAV ja : (A, B) c (Ux,VA) satisfy µiau = As o their limit j* = lim jA : lim HQ(Ua, VA) A

H9(A, B)

A

is defined. Recall that X is paracompact if it is Hausdorff and every open cover has an open locally finite refinement. The proof of the following continuity property of the Alexander-Spanier cohomology theory in paracompact spaces can be found in Spanier [126].

Proposition 2.7. If X is paracompact and A and B are closed, then j* is an isomorphism for all q. Recall that X is locally contractible if for every x e X and neighborhood

U of x, there is a neighborhood V c U of x that is contractible in U. If X is paracompact and locally contractible, then there is an isomorphism p from Hq(X) to the singular cohomology group H%(X) with Z2-coefficients (see, e.g., Spanier [126]). Finally we recall that there is also an isomorphism h from Hs (X) to the vector space Hom(Hq(X), Z2) of linear maps from the singular homology group Hq(X) to Z2 since our coefficient group Z2 is a field.

2.4. Principal Z2-Bundles We briefly recall the notions of paracompact Z2-spaces, principal Z2-bundles, and classifying maps. A basic reference is Steenrod (127). Writing the group Z2 multiplicatively as {1, -1}, a paracompact Z2-space is a paracompact space X together with a mapping µ : 7G2 x X X, called a Z2-action on X, such that

µ(1,x) = x, -(-x) = x Vx e X where -x := p(-1, x). The action is fixed-point free if

-x#x A subset A of a paracompact Z2-space X is invariant if

-A:={-x:xA}=A,

2.4. PRINCIPAL Z2-BUNDLES

and a map f : X

35

X' between two such spaces is equivariant if

f (-x) = - f (x) Vx e X. Two spaces X and X' are equivalent if there is an equivariant homeomorphism f : X X'. We denote by F the set of paracompact free Z2-spaces, identifying equivalent spaces. Example 2.8. Symmetric subsets of normed linear spaces that do not contain the origin are in F, and odd maps between them are equivariant. In -x is particular, S'a-1 e F and the antipodal map S` Sn-1, x equivariant. A principal 7G2-bundle with paracompact base is a triple _ (E, p, B) consisting of an E e F, called the total space, a paracompact space B, called the base space, and a map p : E B, called the bundle projection, such

that there are (i) an open covering {Ua}AEA of B,

(ii) for each A e A, a homeomorphism VA : Ua X Z2 p 1(Ua) satisfying

pa(b, -1) _ -pa(b, 1), pya(b, ±1) = b Vb e B. Then each p1(b), called a fiber, is some pair {e, -e}, c e E. A bundle map f : -> l;' consists of an equivariant map f : E a map f : B if such that p'f = fp, i.e., the diagram

E

f

P1

E' and

El

p!

f B' commutes. Two bundles e and {' are equivalent if there are bundle maps f : 6 - 1' and f' : 6' e such that f' f and f f' are the identity bundle maps on 6 and l;', respectively. We denote by Prinz2 B the set of princiB

pal 7G2-bundles over B and Prin 7G2 the set of all principal 7G2-bundles with paracompact base, identifying equivalent bundles.

Each paracompact free Z2-space can now be identified with a principal 7G2-bundle with paracompact base as follows. Let X = X/Z2 be the quotient space of X E= F with each x and -x identified, called the orbit space of X,

and it : X -> X the quotient map. Then

P:F-'PrinZ2i X--'(X,2r,X) is a one-to-one correspondence identifying F with Prin Z2.

Example 2.9. P(S"-1) = (S'"-1, rr, RP's-1) where 7r identifies antipodal points.

:

Srz-I

.

RP"-1

2. BACKGROUND MATERIAL

36

A map f : B -* B' induces a bundle f

(f * (E'), p, B) e Prin Z2,

called the pullback, where

f*(E')={(b,e')eBxE': f(b)=p'(e)}, -(b,e')=(b,-e') and

Then f f:f

f * (E')

p(b,e)=b. E', (b, e') e' and f constitute a bundle map

i.e., the diagram

f*(El) PI

E'

pl

B f B' commutes. Homotopic maps induce equivalent bundles. Principal Z2-bundles over a given paracompact space B, and hence also paracompact free Z2-spaces via P, can now be classified as follows. For each B' e PrinZ2, we have the mapping

'1: [B, B'] 2 Prinz2 B,

[f] -' f *

where [B, B'] is the set of homotopy classes of maps from B to V.

Proposition 2.10 (Dold [45]). For the bundle ' = (S°D,7r,RP ), called the universal principalZ2-bundle, T is a one-to-one correspondence.

So all principal Z2-bundles with paracompact base are obtainable as pullbacks of the universal bundle (S°, 7r, ]RP°), and principal Z2-bundles over B are classified by homotopy classes of maps from B to the base space 1RP° of the universal bundle, called the classifying space. Thus for each X e F, there is a map f : X -> RPCO, unique up to homotopy and called the classifying map, such that

T([fl) = P(X). 2.5. Cohomological Index We recall the construction and some properties of the cobomological index of Fadell and Rabinowitz [49] for Z2-actions. This index is defined for each paracompact Z2-space, but we will not have occasion to use non-free actions in this text, and therefore will define the index only for free spaces. Let f RPW be the classifying map of X e .F and f* : H*(1[8P®)

H*(X)

the induced homomorphism of the cohomology rings. Referring to Example 2.6, the Z2-cohomological index of X is defined by

i(X) = sup {k > 1 : f*(wk-1) # 01,

2.5. COHOMOLOGICAL INDEX

37

which is well-defined since homotopic maps induce the same homomorphism

by (ci) Taking we = 1 e He(1RPW), f*(wo) = 1 e H°(X) and hence i(X) > 1 when X 0 0. We set i(Z) = 0. Example 2.11. The classifying map of Sn-1 is the inclusion RP"-I c 1l P",

which induces isomorphisms on Hq for q C n - 1, so i(S"-I) = n. In particular, we have the normalization i(Z2) = i(S°) = 1, and i(SW) = oo. The following proposition lists the basic properties of the index.

Proposition 2.12 (Fadell-Rabinowitz [49]). The index i : F

N u {0, oo}

has the following properties.

(ii) Definiteness: i(X) = 0 if and only if X = 0. (i2) Monotonicity: If f : X Y is an equivariant map (in particular, if X c Y), then i(X) S i(Y). Thus, equality holds when f is an equivariant homeomorphism. (i3) Dimension: If X is a symmetric subset of a norrned linear space W and does not contain the origin, then

i(X) 6 dim W.

(i4) Continuity: If X e F and A is a closed invariant subset of X, then there is a closed invariant neighborhood N of A in X such that

i(N) = i(A). When X is a metric space and A is compact, N may be chosen to be a 6-neighborhood N5(A).

(i5) Subadditivity: If X e F and A and B are closed invariant subsets of X such that X = A u B, then

i(A u B) S i(A) + i(B). (is) Stability: If SX is the suspension of X # 0, then

i(SX) = i(X) + 1. (i7) Piercing property: If X, A E F, X0 and XI are closed invariant X is subsets of X such that X = X0 u X1, and ep : A x [0,1] a continuous mapping satisfying y(-x, t) = -V(x, t) for all (x, t), yp(Ax [0, 1]) is closed in X, w(Ax {0}) c X0, and cp(Ax {1}) c XI, then i(p(A x [0, 1]) n Xe n XI) i(A).

(i8) Neighborhood of zero: If U is a bounded closed symmetric neighborhood of the origin in a normed linear space W, then i(dU) = dim W.

2. BACKGROUND MATERIAL

38

PROOF. (ii) This is clear from the definition of the index. (i2) We have the commutative diagram Hk-1(RP-0) 9*1

9*1

Hk-1(Y)

Hk-1(Y)

¶*

f*1

f*

Hk 1(X) Hk-1(X) where g is the classifying map of Y and 7 : X Y is the induced map, with gf serving as a classifying map for X. Since f*g*(wk-1) # 0 implies g*(wk-1) # 0, the conclusion follows. (i3) The radial projection onto the unit sphere, S,

:

f

X ~

x

114

is odd, so

i(X). S i(S) = dim W by (i2) and Example 2.11. (14) If k := i(A) = oo, then by (i2), we can take N = X, or N5(A) with any S > 0 when X is a metric space, so we assume that k < oo.

The set A of all neighborhoods of A in X is directed downward by inclusion. If U is a neighborhood of A, there is a closed neighborhood

V c U of A since X is normal and A is closed, and then V n -V c U is a closed invariant neighborhood of A since A is invariant, so the subset N of closed invariant neighborhoods Na is cofinal in A. Thus, by Proposition 2.7,

j* = lim7a : lmHk(N),) ^ Hk(A) N N where ja : A c NA. Now, we have the commutative diagram -* aA

Hk(N,)

N

f* Hk (RPCO)

lim H (N) A

(fAA )*

Hk(A)

where fA is the classifying map of Na, with faj,, serving as a classifying map for A. Since i(A) = k, zafa(wk) _ (fxaa)*(wk) = 0,

and since j* is an isomorphism, then iafa(wk) = 0. Proposition 2.2 now gives a p e N such that (fa2Aµ)*(wk) = 2AµfA(wk) = 0

2.5. COHOMOLOGICAL INDEX

39

where iaµ : Nµ c NA, with faixN, serving as a classifying map for Nµ, so i(N,1) < k. The desired conclusion follows since k i(N,1) by (i2). When X is a metric space and A is compact, b := dist(A, Nµ) > 0, and i(N5(A)) = k since A c Na(A) c NN,. (i5) Clearly, we may assume that k := i(A) and l := i(B) are finite. By (i,I), there are closed invariant neighborhoods M of A and N of B in X such

that i(M) = k,

i(N) = 1,

and we have the commutative diagram

Hk(X, M)

A*

I Hk(X)

f*

'

Hk(M)

(fo*

Hk(RPc) where the top row comes from the exact sequence of the pair (X, M) and f is the classifying map of X, with f i I serving as a classifying map for M. Since i(M) = k, iif*(wk) _ (fiI)*(wk) = 0,

so f*(wk) e keril = j1(Hk(X,M)) by exactness, say, f*(wk) = jra1. Similarly, f *(wt) = j2 a2 for some a2 E HI (X, N) where j2 : X C (X, N). Now, since the interiors of M and N cover X, we have the commutative diagram Hk+l(X, M u N) Hk(X, M) Ix HI (X, N)

i xi

i*1 H"(X) x HI(X) Hk+1(X) where the horizontal maps are cup products and j c (X, M u N). Since 1

MuN=X and hence H*(X,Mu N) = 0, f*(wk+l)

= f *(wk " WI) = f*(wk) .. f*(WI)

al "j2 a2 =i*(al' a2)=i*0=0, *

= .71

soi(X)
that k < oc. Let A = X x [-1/2,1/2], B+ = (X x [1/2, 1])/(X x {1}), B- = (X x [-1, -1/2])/(X x {-1}), and B = B+ u B-. Then A and B are closed invariant subsets of SX such that SX = AuB, so i(SX) < i(A)+i(B) by (i5). Since the projection onto the first factor Pr1 : A X, (x, t) -* x is equivariant, i(A) 5 k, and i(B) = 1 by Proposition 2.14 (i) below (whose proof uses only (i2)), so i(SX) 5 k + 1. Thus,

k = i(X) S i(SX) 5 k + 1.

2. BACKGROUND MATERIAL

40

If i(SX) = k, then by Proposition 2.14 (iv) below (whose proof is independent of (io)), the rank of io : Hk-1(SX) Hl-'(X) is at least 1 + 8k1, contradicting the fact that io is homotopic to the constant map i1 : X SX, x (x, 1) via the imbedding F : X x [0, 1] SX, (x, t) -. (x, t) and hence

rankio* = rankii = 6k1. So i(SX) = k + 1. (i7) Let Yo = cp-1(Xo), Y1 = y1(X1), and Y = A x [0,1]. Then Yo and Y1 are closed invariant subsets of Y such that Y = Yo u Y1. Since y maps B = Yo nY1 into C = co(Y) n Xo n X1i (i2) gives i(C) > i(B). We will show that the map f = Pr1I.: B -> A, where Pr1 : Y A is the projection onto the first factor, induces injections f* : H"(A) HQ(B) for all q and hence i(B) > i(A). Since the retraction r = Pr1lyo Yo A induces injections r*, it suffices to show that io : B c Yo induces injections io.

f

H°(A) r H(Yo) H0(B)

We have the inclusions Yo io

A x {0}

ko

to

7o

B

Y Y1

k

Ax{1}

and the Mayer-Vietoris exact sequence

Hq(y) '- HQ(Y0)@H,(Y1)

`2

HQ(B)

where j* = (j0, -jr) and P = io + ii (see, e.g., Spanier [1261). If ioa = 0,

*(a,0)=ioa+210=0, so (a,0) e keri* = j*(H4(Y)) by exactness, say, (a,0) =.I *f3 = (jof3,-310). Then jif = 0 and hence 110 = kiji/3 = 0, which implies that ,Q = 0 since l1, and hence also 11, is a homotopy equivalence, so a = joQ = 0. (is) Taking R > 0 so large that BR(0) D U and applying Corollary 2.13 below (whose proof uses only (i2) and (i7)) to the identity map on ISR(O) = BR(0) gives

i(W) = i(BR(0) n aU) > i(SR(0)) = i(S) = dimes, and (i3) gives the opposite inequality.

2.5. COHOMOLOGICAL INDEX

41

Corollary 2.13. If U is a bounded closed symmetric neighborhood of the origin in a nonmed linear space W, A is a bounded symmetric subset of U, and 4 : IA W is an odd continuous mapping such that vl)(IA) is closed and ?IA = idA, then i(v/)(IA) n SU) > i(A). PROOF. We have z/ (IA) n SU = zl (b-1(aU)), so

i(v/5(IA) n 8U) > i(r-1(5U)) by (i2). Since ?G is odd, v/b(0) = 0, so there is a 6 > 0 such that O

V6:= {tx : x e A, t E [0, 6)} c v)-'(U) by continuity. We apply (i7) to o : A x [0,1] IA\V4 = (vG-1(U)\Vs) u (2.3)

(x,t)'-* ((1 -t)6+t)x. Since V5 is contained in the closed set ip-1(U), so is its relative boundary dV4, so yp(A x {0}) = 5714 c v,-1(U)\V4. Since v/' is the identity on A c U',

o(A x {1}) = A c 0-1(U). Since p is onto, w(A x [0,1]) n (,G-1(U)\V5) n 0-1(U) _-1(5U)\14 = v/ -'(5U) by (2.3), so we have i(v/)-1(57U)) > i(A).

The following proposition gives some additional properties of the index.

Part (iii) is due to Degiovanni and Lancelotti [43], and (iv) to Cingolani and Degiovanni [30].

Proposition 2.14. Let X E .F with index k > 1. (i) If X is the disjoint union of a pair of subsets U, -U, then k = 1.

In particular, k = 1 when X is a finite set. (ii) If X is compact, then k < oo.

(iii) If X is locally contractible, then for each finite j < k, X has a compact invariant subset C with i(C) >, j. In particular, there is a compact invariant subset C with index k when k < oo. (iv) If k < oo and A is an invariant subset of X with index k, then the rank of i* : Hk-1(X) , Hi`-1(A), induced by i : A c X, is at least 1 + 6k I. In particular, Hk-1(X) # 0. PROOF. (i) The map

f:X-S°,

x--.

1,

-1, x E -U

is equivariant, so

1(Ic i(S°)=1 by (i2) and Example 2.11.

xEU

2. BACKGROUND MATERIAL

42

(ii) Each x e X has a closed neighborhood U. such that Ux n -Ux = 0 since x # -x and X is regular, and a finite number of them, Uxl, ... , U, cover X by compactness. Then m

i(X)<'i(Uxjv-Ux,)=in j=1

by (i5) and (i). (iii) For any invariant subset C of X, we have the commutative diagram Hj-1(RP0D)

µ

H,'4-'(1RP')

h

Hom(Hj_1(RPC),Z2)

f, l

A

(f*)* 1

HS-1(X)

h

Hs-1(C)

h

_

Hom(Ha_1(X),7G2)

Hom(Hj_1(C),7G2)

where f is the classifying map of X and i : C c X is induced by i : C c X, with f i serving as a classifying map for C. Starting from the top left-hand corner, f*(wj-1) # 0 since j < k, t is an isomorphism since X is locally contractible, and h is also an isomorphism, so the map a:= h. µ f * (wj-1) # 0 in Hom(Hj_1(X), Z2), say, a(z) # 0, z e Hj_1(X). Since singular homology is a theory with compact supports, X has a compact invariant subset C such

that zisintheimage ofi*:Hj_1(C)

Hj_1(X),say, z=ti*c, ceHj_1(C).

Then from the lower part of the diagram (h.µi.* f*(wj 1))(c) _ ((2*)*a)(c) = a(@*c) = a(z) and hence (fi)*(wj-1) = i*f*(wi-1) # 0, so i(C) (iv) We have the commutative diagram

Hk-1(RP`O) 'r* f*1 Hk-1(X)

Hk-1(S`°)

f*1 'r*

n*

j.

Hk-1(RPw)

Hk(]RPoO)

f*

f*1

Hk-1(X) r* Hk-1(X) *1

I

r*

0,

"f*w

Hk(X) a*1

i*1 "i*f*W

Hk-1(A) Hk(A) r* Hk-1(A) Hk-I(A) where f is the classifying map of X, i : A c X is induced by i : A c X, with fi serving as a classifying map for A, and the horizontal rows come from the Thom-Gysin exact sequence (see, e.g., Spanier [126]). Since i(X) = k, f*(wk-1) " f*w = f*(wk-1 " w) = f*(wk) = 0,

2.5. COHOMOLOGICAL INDEX

43

so f*(Wk-1) e ker('- f*w) = r*(Hk-1(X)) by exactness, say, r*a. Then from the lower middle square r*i*a = 8*r*a = 2*f*(Wk 1)

_

(fi)*(Wk

1)

f*(Wk-1) =

0-0

since i(A) = k, so i*a # 0. When k = 1, i*a # 7r*1 = 1 since i*a 0 kerr* = 7r*(H0(A)), and i*1 = 1. The second assertion follows by taking A = X.

CHAPTER 3

Critical Point Theory In this chapter we develop the critical point theory needed for our treatment of equation (1.1). Let 4) be a real-valued function defined on a real Banach space W of dimension d >, 1. We say that D is Frechet differentiable at u c- W if there is an element 4)'(u) of the dual W*, called the Frechet derivative of (D at it, such that

4)(u + v) = D(u) + (4)'(u), v) + o(jjvIj) as v 0 in W is the duality pairing. The functional D is continuously Frdchet where differentiable on W, or belongs to the class C1(W, R), if 4)' is defined everywhere and the map W 4)'(u) is continuous. We assume that W*, it 4) e C' (W, IR) for the rest of this chapter. Replacing 4) with D - 4)(0) if necessary, we may also assume that 4)(0) = 0. We say that it is a critical point of 4) if 4)'(u) = 0. A real number c is a critical value of $ if there is a critical point it with $(u) = c, otherwise it is a regular value (in particular, any c that is not a value of ID is a regular value!). We use the notations

a = {u e W : 4)(u) >, a},

fib

=

{u c W : 4)(u) < b},

-Da=Da n 0b

K= {u e W: 4)'(u) = 0},

W= W\K1

K`=Kc

=Kn(Da,

for the various superlevel, sublevel, critical, and regular sets of 4).

The functional 4) is called even if it is invariant under the antipodal action of 7L2, i.e.,

4'(-u)

4)(u)

Vu E W.

Then (' is odd, i.e.,

V(-'u) _ -4)'(u)

Vu E W.

We denote by

=1} the unit sphere in W and 7Fs : W\ {0}

S,

a lluull

45

46

3. CRITICAL POINT THEORY

the radial projection onto S.

3.1. Compactness Conditions It is usually necessary to have some "compactness" when seeking critical points of a functional. The following condition was originally introduced by Palais and Smale [91].

Definition 3.1. 4) satisfies the Palais-Smale compactness condition at the level c, or (PS)0 for short, if every sequence (uj) c W such that 1k(uj) - c, 4)'(u1) -> 0, called a (PS)0 sequence, has a convergent subsequence. 4) satisfies (PS) if it satisfies (PS)0 for every c e IR, or equivalently, if every sequence such that (4)(uj)) is bounded, V'(uj) -. 0, called a (PS) sequence, has a convergent subsequence. The following weaker version was introduced by Cerami [25]. Definition 3.2. 1) satisfies the Cerami condition at the level c, or (C)0 for short, if every sequence such that '(uj) --* c, (1 + IIujII)4)'(ui) -0, called a (C),, sequence, has a convergent subsequence. 4) satisfies (C) if it satisfies (C)0 for every c, or equivalently, if every sequence such that (,P(nj)) is bounded, (1 + mujll) 4)'(uj) 0, called a (C) sequence, has a convergent subsequence. This condition is weaker since a (C)0 (resp. (C)) sequence is clearly a (PS)0 (resp. (PS)) sequence also. Note that the limit of a (PS)0 (resp. (PS)) sequence is in K` (resp. K) since 4) is C'. Since any sequence in K` is a (C)0 sequence, it follows that K° is compact when (C)0 holds. It suffices to show that (uj) is bounded when verifying these conditions for the functional in (1.4) by Lemma 3.3. If 4) is as in (1.4), then every bounded sequence (uj) such that 4)'(uj) -* 0 has a convergent subsequence. PROOF. A renamed subsequence converges weakly to some it since W is reflexive, and f (uj) converges in W* for a further subsequence since f is compact. Then (Ap uj, uj - u) _ (41'(uj) + f (uj), uj - u) - 0 by Lemma 3.4 below, so uj -* it for a subsequence by (A4).

Lemma 3.4. If (Lj) c W* converges to L and (uj) c W converges weakly to u, then (Lj, uj) -> (L, u), in particular, (Lj, uj - u) 0. PROOF. Since (uj) is bounded by the principle of uniform boundedness,

I(Lj,uj)-(L,u)Is1ILj-LII*11ujll+I(L,uj)-(L,u)I-*0.

3.2. DEFORMATION LEMMAS

47

3.2. Deformation Lemmas Deformation lemmas allow to lower sublevel sets of a functional, away from its critical set, and are an essential tool for locating critical points. The main ingredient in their proofs is usually a suitable negative pseudo-gradient flow, a notion due to Palais [93].

Definition 3.5. A pseudo-gradient vector field for 4) on W is a locally Lipschitz continuous mapping V : W W satisfying

V(v)

(3.1)

(114)f(u)11*)2

V(u)2 (V(u), V(u))

Vu C W.

Lemma 3.6. There is a pseudo-gradient vector field V for 4' on W. When 4' is even, V may be chosen to be odd. PROOF. For each u e W, there is a w(u) e W satisfying IIw(u)II < 11V(u)11*

,

2 (4 (u), w(u)) > (b (u)II*)2

by the definition of the norm in W*. Since 4V is continuous, then (3.2)

hw(u)11 _ D

(v)11* ,

2 (4'(v),w(u)) > (l p,(v)11*)2

Vv e Nu

for some open neighborhood Nu c W of it. Since W is a metric space and hence paracompact, the open covering {Nu}uEw has a locally finite refinement, i.e., an open covering {Na},EA of W such that (i) each Na c Nua for some ua e W, (ii) each u e W has a neighborhood U. that intersects Na only for A in some finite subset A. of A (see, e.g., Kelley [59]). Let {cpa},,eA be a Lipschitz continuous partition of unity subordinate to {Na},EA, i.e., (i) spa e Lip (W, [0,1]) vanishes outside Na, (ii) for each it e W, Z cpa(u) = 1

(3.3)

.SEA

where the sum is actually over a subset of Au, for example, dist(u, W\Na)

a

(u)

' dist (u,

W\N,).

AeA

Now V(u)=

' Wa(u)w(ua) AeA

is Lipschitz in each Uu and satisfies (3.1) by (3.2) and (3.3).

3. CRITICAL POINT THEORY

48

When $ is even, V is odd and hence -V(-u) is also a pseudo-gradient, and therefore so is the odd convex combination (V(u) - V(-u)). The following deformation lemma improves that of Cerami [25].

Lemma 3.7 (First Deformation Lemma). If c e 1@, C is a bounded set containing ICS, 5,k > 0, and 4) satisfies (C)0, then there are an eo > 0 and,

for each e e (0,so), a map g e C([0,1] x W, W) satisfying

(i) q(0,) = idw, (ii) rl(t,) is a homeomorphism of W for all t e [0, 1], (iii) ?7(t,) is the identity outside A = <,+2E\Nbi3(C) for all t e [0, 1], (iv) 11 ?7(t, n) - nil < (1 + hull) 8/k V(t, u) e [0, 1] x W, (v) u)) is nonincreasing for all u e W,

(vi) i7(1, P"e\N6(C)) C 4D`-£. When
for all te[0,1]. First we prove a lemma.

Lemma 3.8. If c e R, N is an open neighborhood of K0, k > 0, and 4) satisfies (C)0, then there is an eo > 0 such that inf

(1 + hull)Dk e

Ve e (O,eo).

UE$ +£\N

PROOF. If not, there are sequences ei

0 and uj e
(1 + jIujII) 114''(u.i)< key. Then (uj) c W\N is a (C)0 sequence and hence has a subsequence converging to some u e K0\N = 0, a contradiction. PROOF OF LEMMA 3.7. Taking k larger if necessary, we may assume

that (3.4)

(1 + Dull )/k < 1/3 Vu E N5(C).

By Lemma 3.8, there is an eo > 0 such that for each e e (0, eo), (3.5)

(1 + IIu1I) IV (u)ll*

>-

Vu e A.

ln(1 + b/k) Let V be a pseudo-gradient vector field for
g(u) =

dist(u, A) + dist(u, B)' and r1(t, u), 0 S t < T(u) < x the maximal solution of (3.6)

_ -4eg('7)

lil ((1))I

t > 0, 2

71(0, u) = u e W.

3.2. DEFORMATION LEMMAS

49

For 0 <s < t < T(u), 11,q (t, u) - q (s, u) 11 <4E f

llV(n(T,u))II

dr

ft

< 8e

g(71(r,u))

ln(1 + 6/k)

by (3.5)

(1 + Ilrl(T, u) II ) dT

Js

ln(1 + 6/k) [

by (3.1)

dT

J

11 7) (T, u)

- 71(8,u) II dT

+(1+1177(s,u)11)(t-s)J, and integrating gives

(3.7)

II,l(t,u)-'7(s,u)II

(1+11,q(S,u)II)((1+6/k)t-s-1).

Taking s = 0 we see that 1177(, u) II is bounded if T (u) < co, so T(u) = oo and

(i) - (iv) follow. By (3.6) and (3.1), (3.8)

at

(143(77(t,u)))

-dsg(rl)

(4)II(

V

2s g(r7) < 0

and hence (v) holds. To see that (vi) holds, let u e is+6\Ns(C) and suppose that 71(1, u) 0 4 Then r7(t, u) e 4)',+E for all t e [0, 1], and we claim that 77 (t, u) 0 N2613 (C) . Otherwise there are 0 < s < t < 1 such that dist(i7(s, u), C) = 6,

26/3 < dist(r7(r, u), C) < 6,

T e (s, t),

dist(i7(t, u), C) = 26/3.

But, then 6/3 < 11o(t, u) - 71 (s, u)II < (1 + 11i (s, u)II ) 6/k < 6/3

by (3.7) and (3.4), a contradiction. Thus, ?7(t, u) e B and hence g(t7(t, u)) _ 1 for all t e [0,1], so (3.8) gives 4 (?7 (1, u)) < 4' (u) - 2s < c - e,

a, contradiction. When (D is even and C is symmetric, A and B are symmetric and hence g is even, so 77 may be chosen to be odd in it by choosing V to be odd.

Lemma 3.7 provides a local deformation near a (possibly critical) level set of a functional. The following lemma shows that the bomotopy type of sublevel sets can change only when crossing a critical level.

3. CRITICAL POINT THEORY

50

Lemma 3.9 (Second Deformation Lemma). If -co < a < b <.+oo and 4b has only a finite number of critical points at the level a; has no critical values

in (a, b), and satisfies (C), for all c e [a, b] n IR, then 4Q is a deformation retract of $b\Kb.

PROOF. Let V be a pseudo-gradient vector field for 4) and ((t,u) the solution of (3.9)

-IIV(0)112,

t> 0,

((0,u) = u e (Db\(V u Kb) c IV.

Then d

dt

(ID'((), V(())

(W(t,u))) _

<

IIV(()II2

1

2

by (3.1) and hence (3.10)

(((t, u)) < (((s, u))

s),

2 (t

0(8< t.

Taking s = 0 and using 4P(((0, u)) _ 4D(u) < b gives a T(u) e (0, 2(b - a)] such that (b(((t, u)) \, a as t J' T(u). We set T (u) = 0 and ((0, u) = u for

ueV.

For b > 0, let

T = {t e [0,T(u)) : dist(((t, u), Ka) > E}. Then rna := inf (1 + II((t, u)II) D tET

(((t, u))jI3 > 0

by (C), so (3.11)

u) - ((s, u) 11 <

f

t

ft

<2J 2

by (3.9)

IIV(((T u))II dT

III;(((T,u))II ft (1+II((T,u)II)dr,

by (3.1)

[s,t]c7a.

Case 1: (( , u) is bounded away from K'. Then T = [0,T(u)) for some S > 0 and II((', u) 11 is bounded as in the proof of Lemma 3.7, so (3.12)

II((t, u) - ((s, u) 11 < C (t - s),

0

s < t
for some C > 0. Let tj / T(u). Taking t = tj and s = tk in (3.12) shows that (((tj,u)) is a Cauchy sequence and hence converges to some v c 4D-r(a)\K°. Now taking s = tj shows that ((t, u) v as t / T(u). We set ((T(u), u) = v and note that T is continuous in it in this case. Case 2: ((., u) is not bounded away from K°. We claim that then ((t, u) converges to some v e K° as t / T (u). Since Ka is a finite set, otherwise

3.2. DEFORMATION LEMMAS

51

there are a vo e K 6 > 0 such that the ball B3b(vo) contains no other

points of K', and sequences sj, tj / T(u), sj < tj such that 6 < 11((T, u)

S(sj,u) - v0l = b,

- vo11

< 26,

T e (sj,tj),

11((tj, u) - vo 11 = 26.

But, then

as

sc(tj-sj)-.0

by (3.11), a contradiction.

We set ((T(u), u) = v and claim that T is continuous in it in this case also. To see this, suppose that uj -> it. We will show that

T:=limT(uj)>, T(u) >, limT(uj)=:T and hence T(uj)

T(u).

If T < T(u), then passing to a subsequence,

((T(uj),uj) -* (T, u) and hence a = 4(((T(uj),uj)) - ID(((!:, u)) > a, a contradiction. If T > T(u), then for a subsequence and any t < T(u),

a = D(((T(uj), uj)) S fi(((t, uj)) -

2

(T(uj) - t)

by (3.10), and passing to the limit gives

a < P (((t,u)) - 2 (- t) again a contradiction. We will show that

a - 2 (T -T(u)) < a as t / T(u),

is continuous. Then

rl(u, t) = ((tT(u), u) will be a deformation retraction of (pb\Kb onto $a. Case 1: it e 4Db\(.D° u Kb), 0 < t < T(u). Then (is continuous at (t, u) by standard ODE theory. Case 2: it e Db\(4' u K6), t = T(u). Suppose that uj e 4)b\(pa U

K"), 0 -< tj <- T(uj), (tj, uj) (T(u), u), but ((tj, uj) -f. ((T(u), u) =: v. Then there is a S > 0 such that (B3b(v)\ {v}) n K' _ 0 and (3.13)

11((tj, uj) - v11 -> 2b

for a subsequence. Since ((s, u) converges to v as s J' T (u), (3.14)

11((s,u)-v11 <- 6/2

for all s < T(u) sufficiently close to T(u). For each such s, (3.15)

11((s,uj) - ((s, u)11 -< 6/2

for all sufficiently large j by Case 1.

Taking a sequence sj / T(u) and

combining (3.14) and (3.15) gives tj > sj and (3.16)

11((sj,uj) -v11 s 5

3. CRITICAL POINT THEORY

52

for a further subsequence of (ti,uj). By (3.13) and (3.16), there are se-

s

quences (3.17) 1

tj < tj such that

(sj,uj)-vll =6,

<

IIK(T,uj) - vll < 26,

II((tj,'aj)

- vll

7- c- (s', tj)

= 26.

Then

SCD((tj,nj)-((sj,uj) CC(tj-s'.)-.0

(3.18)

by (3.11), a contradiction.

Case 3: u c D-'(a), t = 0. Suppose that uj e $6\K6, 0 < tj < T(uj), (tj,uj) -> (0, u), but ((tj,uj) -» ((0,u) = u. Then there is a 6 > 0 such that (B36(u)\ {u}) n K° _ 0 and

II((tj,uj)-ull X26

(3.19)

for a subsequence. Since ((0, uj) = uj -* u,

II((0,uj) - ull < 6

(3.20)

for sufficiently large j. By (3.19) and (3.20), uj E 4pb\(,pd u K6), tj > 0, and there are sequences 0 < s' < tj < tj for which (3.17), and hence also (3.18), holds. Case 4: u. e

t = 0. Then ((0, u) = u.

3.3. Minimax Principle First deformation lemma implies that if c is a regular value and (P satisfies (C)C, then the family D, E of maps 77 E C([0, 1] x TV, W) satisfying (i) 77(0, .) = idw,

(ii) r7(t, ) is a homeomorphism of W for all t c [0,1], (iii) 77(t,) is the identity outside DI+26 for all t e [0,1], (iv) 4 u)) is nonincreasing for all u e W, (v)

77(1,

Dc+e)

C P`

is nonempty for all sufficiently small e > 0. We say that a family F of subsets of W is invariant under E),,, if

MaT,77ED,,E

s7(1,M)EF.

Proposition 3.10 (Minimax Principle). If F is a family of subsets of W. (3.21)

c := inf sup (D(u) MeT nEAt

is finite. F is invariant under DC,e for all sufficiently small e > 0, and 4" satisfies (C)C, then c is a critical value of D. PROOF. If not, taking e > 0 sufficiently small, M E F with sup 4 (M) < c+ e, and 7) E D..E, we have r7(1, M) E F and sup q )(77(l, M)) < c - e by (v), contradicting (3.21).

3.4. CRITICAL GROUPS

53

We say that a family r of continuous maps 7 from a topological space X into W is invariant under D0, E if

yEP,rleD,,E

q(i,-)o-Ycr.

Minimax principle is often applied in the following form.

Proposition 3.11 (Minimax Principle). If P is a family of continuous maps y from a topological space X into W, c:= inf sup 4)(u) ryeP uE'Y(X)

is finite, r is invariant under I3. e for all sufficiently small e > 0, and 4) satisfies (C)c, then c is a critical value of D. PROOF. Apply Proposition 3.10 with .F = {y(X) : -Y C- 171,

0

3.4. Critical Groups In Morse theory the local behavior of D near an isolated critical point u is described by the sequence of critical groups Cq((p, u) := Hq((D° n U, 4)` n U\ {u}), q > 0 (3.22) where c = 4)(u) is the corresponding critical value and U is a neighborhood of u containing no other critical points of 4). They are independent of U, and hence well-defined, by (c3). Critical groups help distinguish between different types of critical points and are extremely useful for obtaining multiple critical points of a functional (see, e.g., Chang [28)). One of the consequences of the second deformation lemma is the following proposition relating the change in the topology of sublevel sets across a critical level to the critical groups of the critical points at that level.

Proposition 3.12. If -co < a < b < +cc and 4) has only a finite number of critical points at the level c E (a, b), has no other critical values in [a, b], and satisfies (C)e for all c' E [a, b] n R, then

Hq(,D6,,1)a). O Cq(4),u)

Vq.

ueK0

In particular,

dimHq(4)b,(ba) =

)' dimCq(4),u)

Vq.

uEK°

PROOF. We have Hq(4)b, (Da) , Hq(4)`,'Fa) x Hq(4)`, 4)'\K`) (3.23)

since 4)` and 4)d are deformation retracts of 4)b and 4)`\K°, respectively, by Lemma 3.9. Taking 6 > 0 so small that the balls B5(u), U E K` are mutually disjoint and then excising D`\ UuEK° B5(u), we see that the last group in (3.23) is isomorphic to O Hq(4)` n B5(u), 4)` n BS(u)\ {u}) = O Cq(4), u). uEK0

uEK.

3. CRITICAL POINT THEORY

54

For the change in the topology across multiple critical levels, we have

Proposition 3.13. If -oo < a < b < +co are regular values and 4' has only a finite number of critical points in oba and satisfies (C), for all c e [a, b] n R, then

dimH9(4)b,.pa) < )' dimC4($,u)

Vq.

uEKa

In particular, $ has a critical point it with a < $(u) < b and C'(4), u) # 0 when HQ(4b, (pa) # 0.

First we prove a lemma of a purely topological nature. Lemma 3.14. If X1 c ... c Xk+l are topological spaces, then k

dimH9(Xi+1,Xi)

dimH4(Xk+1,X1) <

(3.24)

Vq.

i=1

PROOF. In the exact sequence H4(Xk+1, Xk)

j*

H°(Xk+1,X1) HQ(Xk, XI)

of the triple (Xk+1,Xk,X1), imi*

HQ(Xk+l,Xl)/keri* = H9(Xk+1,X1)/imj*

and hence

dim H4 (Xk+l, Xl) = ranki* + rank j * dim H4 (Xk, X1) + dim H4 (Xk+1, Xk).

Since equality holds in (3.24) when k = 1, the conclusion now follows by induction on k. PROOF OF PROPOSITION 3.13. Let cl < .

< ck be the critical values

in (a,b) and a=al
dimH4((Db, Da)

Y,'

dim i=1 ueK`i

Y,' dimCQ(0,u). uEKa

3.5. MINIMIZERS AND MAXIMIZERS

55

When the critical values are bounded from below and D satisfies (C), the global behavior of P can be described by the critical groups at infinity Cq(4>, oo)

(3.25)

Hq(W, 4°),

q >, 0

where a is less than all critical values. They are independent of a, and hence well-defined, by Lemma 3.9 and the homotopy invariance of the cohomology

groups. Since W is contractible and (V = 0 if and only if 4' is bounded from below, the following proposition is a consequence of Proposition 2.4.

Proposition 3.15. Assume that 4' satisfies (C). (i) If 4' is bounded from below, then Cq(,D, oo)

Sqo 7/2.

(ii) If 4) is unbounded from below, then 00)

Hq-1(ID°)

Vq.

In particular, Co(4', oo) = 0. When studying the existence and multiplicity of critical points of a functional we may often assume without loss of generality that there are only finitely many critical points. The following proposition relating the critical groups of 'F at infinity to those of its (finite) critical points is then immediate from Proposition 3.13 with b = +co.

Proposition 3.16. If 4) satisfies (C), then

dimCq(4',co) <)' dimCq(¢,u)

Vq.

ueK

In particular, k has a critical point u with C? (4), u) # 0 when Cq((g, co) # 0.

3.5. Minimizers and Maximizers In this section we give sufficient conditions for 4) to have a global extremum and compute the critical groups at an isolated local extremum.

Proposition 3.17. If 'F is bounded from below (resp. above) and satisfies (C), for c = inf'F (resp. sup 4)), then 'F has a global minimizer (resp. maximizer).

PROOF. Apply Proposition 3.10 with F = { {u} : u e W} (resp. {W}).

0 Turning to critical groups, we have

Proposition 3.18. If u is an isolated local minimizer (resp. maximizer) of 'F, then C"(`p,u) x bgo7G2 (resp. SgdZ2).

3. CRITICAL POINT THEORY

56

PROOF. Let c = 4(u). If u is an isolated local minimizer, then for sufficiently small r > 0, ID(v) >, c

Vv e Br(a)

and there are no other critical points of $ in Br(u). Since any v e Br(u)

with D (v) = c is then a local minimizer, qD(v) > c,

v e Br(u)\{u},

and hence

C° (,P, u) = Hq(Yn Br(u),,D` n Br(u)\ {u}) = Hq({u} , 0) ^ aqi Z2 by (c4).

If u is an isolated local maximizer, then for small r > 0,

'I(v) < c Vv E Br(u), and hence

Cq('D,u) = H'(Br(u),Br(u)\{u}) ,:

SgdZ2

by Example 2.5.

Combining Propositions 3.15 (i), 3.17, and 3.18 gives

Corollary 3.19. If (P is bounded from below, satisfies (C), and has only a finite number of critical points, then co) a bq0 Z2 and has a global minimizer u with Cq(F, u) x 8q0 Z2

3.6. Homotopical Linking The notion of homotopical linking is useful for obtaining critical points via the minimax principle.

Definition 3.20. Let A be a closed proper subset of a topological space X, g e C(A, W) such that g(A) is closed and bounded, B a nonempty closed subset of W such that dist(g(A), B) > 0, and

r = {ry c C(X, W) : 7(X) is closed, 7'A = g}. We say that (A, g) homotopically links B with respect to X if (3.26)

y(X)nB#0 VyEP.

When g : A c W is the inclusion and X = IA, we will simply say that A homotopically links B.

Proposition 3.21. If (A, g) homotopically links B with respect to X, (3.27)

c:= inf sup

4D(u)

7Er nE7(X)

is finite, sup 4)(g(A)) S inf 4(B), and 4 satisfies (C)., then c > inf D(B) and is a critical value of 4. If c = inf 4) (B), then 4) has a critical point with critical value c on B.

3.6. HOMOTOPICAL LINKING

57

PROOF. By (3.26), c 3 inf inf (F(B), and let 2e < c - sup b(g(A)). Then for any n e D,,, and all t e [0,1], 71(t, .) is the identity on g(A) by (iii) in the definition of D, E, so F is invariant under D,,, and hence Proposition 3.11 applies. Now suppose that c = inf 4D(B) and K` n B = 0. Since K' is compact by (C), and B is closed, dist(K°, B) > 0. Applying Lemma 3.7 to -(D with C = g(A) u K° and 8 < dist(C, B) gives an e > 0 and a homeomorphism it of W such that g is the identity outside D,+2E\Nb/3(C) and

\Ns(C)) c'k+E' in particular, 71 is the identity on g(A) and i >, c+e on 77(B). Then taking ayeFwith 4) c c + son 7(X), we have := 02' eF and hence 7(X) n77(B) = 77(ry(X) n B) # 0 by (3.26), a contradiction. To construct examples of homotopically linking sets, let M be a bounded symmetric subset of W\ {0} radially homeomorphic to the unit sphere S, i.e., the restriction of the radial projection 7rs to M is a homeomorphism. Then the radial projection from W\ {0} onto M is given by (WSI'")-r

7rM =

ors.

Let Ao # 0 and Bo be disjoint closed symmetric subsets of M such that (3.28) i(Ao) = i(M\Bo) < co where i denotes the cohomological index. Proposition 3.22. Then A = RAO homotopically links B = 7r j (Bo) u {0} for any R > 0. PROOF. If not, there is a ry e C(IA, BC) with 71A = id _q. Then SAO

M\Bo,

JrM (-y ((I - t) Ru)), (u, t)

t e [0, 1]

-r,I,I(ry(-(1+t)Ru)), te[-1,0)

is an odd map and hence i(M\Bo) i(SAo) = i(Ao) + 1 by (12) and (i6), contradicting (3.28).

Proposition 3.23. If h e C(CAO, M) is such that h(CAo) is closed in M and hIA0 = idA0, then A = R(IAo u h(CAo)) homotopically links B = rBo

for any0
PROOF. If not, there is a 'y e C(IA, B`) such that -y(IA) is closed and ryIA = idA. Since hIAo = idA0 and AO is symmetric, h extends to an odd map h e C(SAO, M), and since RIAO c A is symmetric, ry extends to an odd map ry e C(IA, BC) where A = Rh(SAo). Since j(IA) n rM c rM\B, (3.29)

i(ry(IA) n rM)

i(rM\B) = i(M\Bo)

3. CRITICAL POINT THEORY

58

by (i2). Noting that and 0 = ry gives (3.30)

idA and applying Corollary 2.13 with U = rIM

i(ry(IA) n rM) > i(A) > i(SA°) = i(Ao) + 1

by (i2) and (is). Together, (3.29) and (3.30) contradict (3.28).

3.7. Cohomological Linking The notion of cohomological linking is useful for obtaining pairs of sublevel sets with nontrivial cohomology and hence critical points with nontrivial critical groups via Proposition 3.13.

Definition 3.24. Let A and B be disjoint nonempty subsets of W. We say that A cohomologically links B in dimension q < co if i* : Eq(B`)

fiq(A),

induced by i : A c B'-, is nontrivial.

Proposition 3.25. If A cohomologically links B in dimension q and (3.31)

1A < a < 4?IB,

then H°(W, (b') = 0 and Hq+1(W, 4?a) # 0. If, in addition, a is less than all critical values and 4' has only a finite number of critical points in 4i-1 (a, oo) and satisfies (C), then C°(4?, oo) = 0, Cq+1(4), co) o 0, and 4i has a critical point u with 4>(u) > a and Cq+1(4 u) # 0. PROOF. We have the commutative diagram

Hq(B`)

Hq(4ia)

Hq(A)

induced by A c r c B`. Since i* # 0, Hq(4?Q) # 0 and hence the conclusions follow from Propositions 2.4 (ii) and 3.16.

To construct an example, let M, 7rM, As, and B° be as in the last section.

Proposition 3.26. Then A = RA° cohomologically links B = irk (Bo) u{0} in dimension q = i(Ao) - 1 for any R > 0. PROOF. We have the commutative diagram

Hq(M\Bo)

Hq(B`)

Hq(Ao)

''

Hq(A)

3.8. NONTRIVIAL CRITICAL POINTS

59

where j : A0 c M\B0 and the vertical maps are isomorphisms induced by M\Bo and the homeomorphism the homotopy equivalence rrM16 : B' nM I A : A -* A0. The conclusion follows since j * # 0 by Proposition 2.14 (iv).

If A cohomologically links B in dimension q and h is a homeomorphism of W, then the following commutative diagram, where j : h(A) c h(B`), shows that h(A) cohomologically links h(B) in dimension q as well. H9(hI(B`))

'*

h* I

HQ(h (A)) h*

Hq(LLLB')

Hq(A) In particular, we have the following generalization of Proposition 3.26.

Proposition 3.27. If h is a homeomorphism of W, then A = h(RAo) cohomologically links B = h(7r - (Bo) u {0}) in dimension q = i(Ao) - 1 for

any R>0.

3.8. Nontrivial Critical Points In many applications D has the trivial critical point u = 0 and we are interested in finding others. We assume that 4) has only a finite number of critical points. The following proposition is useful for obtaining a nontrivial critical point with a nontrivial critical group. Proposition 3.28. Assume that 4? satisfies (C). (i) If Cq(4?, 0) = 0 and C"(-D, oo) # 0 for some q, then ID has a critical point It # 0 with Cq((D, u) # 0. (ii) If Cq(4., 0) # 0 and Cq(4i, oo) = 0 for some q, then 4i has a critical point It # 0 with either 4?(u) < 0 and Cq-1(4?, u) # 0, or 4i(u) > 0 and Cq+'(,Du) # 0. First a purely topological lemma. Lemma 3.29. If X1 c X2 c X3 c X4 are topological spaces, then dimHq-1(X2iXI)+dimHq+1(X4,X3)

dimHq(X3iX2)-dimHq(X4iX1)

Hq.

PROOF. From the exact sequence

...

Hq-1(X2, XI)

HQ(X3, X2)

HI (X3, Xj) of the triple (X3, X2, X1), we have

dim Hq-1(X2, X1) > rank 6 = nullity j * = dim Hq(X3, X2) - rankj* dim Hq(X3, X2) - dim Hq(X3i XI),

3. CRITICAL POINT THEORY

60

and from the exact sequence Hq(X4, X1)

Hq(X3, XI) a

H4+1(X4 X3)

...

of the triple (X4, X3, XI), dim Hq(X3i XI) = rank 6+ nullity b < dim Hq+I (X4, X3) + rank i* < dim Hq+I(X4, X3) + dim Hq(X4, X1), so the conclusion follows. PROOF OF PROPOSITION 3.28. (i) By Proposition 3.16, D has a critical point u with Cq(4), u) # 0 since Cq(4), 00) # 0, and u # 0 since C"(4), 0) = 0.

(ii) Let e > 0 be so small that zero is the only critical value in [-e, e]

and a be less than -e and all critical values. Since dimHq((DE 4)-E) dimCq(4), 0) by Proposition 3.12 and Hq(W, 4)n) = Cq(4), co), applying Lemma 3.29 to 4a c $-E c V c W gives dim Hq-I ((I)-E, 4)")+ dim Hq+1 (W, 4p6)

dim Cq(1, 0) - dim Cq(4), oo) > 0.

Then either

Hq-1($-E 41) # 0,

or Hq+I(W, V)

0, and the conclusion

follows from Proposition 3.13.

Remark 3.30. The alternative in Proposition 3.28 (ii) and Lemma 3.29 were proved by Perera [95, 96].

3.9. Mountain Pass Points A critical point it of 4) with C' (4), u) # 0 is called a mountain pass point. Since cohomology groups, and hence also critical groups, are trivial in negative dimensions, the special case q = 0 of Proposition 3.28 (ii) reduces to

Corollary 3.31. If C°((D, 0) # 0, C°(4), oo) = 0, and 4) satisfies (C), then 4) has a mountain pass point it # 0 with D(u) > 0. This implies the well-known mountain pass lemma of Ambrosetti and Rabinowitz [7]. Indeed, if the origin is a local minimizer and 4) is unbounded

from below, then C°(4), 0) x Z2 by Proposition 3.18 and C°(4), co) = 0 by Proposition 3.15 (ii), so Corollary 3.31 gives a positive mountain pass level.

3.10. Three Critical Points Theorem Another consequence of Proposition 3.28 (ii) is

Corollary 3.32. If Cq(4), 0) # 0 for some q > 1 and 4) is bounded from below and satisfies (C), then D has a critical point ui # 0. If q >, 2, then there is a second critical point u2 # 0.

3.11. COHOMOLOGICAL LOCAL SPLITTING

61

PROOF. By Corollary 3.19, Cq(,D, co) = 0 and D has a global minimizer

u1 with Cq(*F, ul) = 0. Since C" ((P, 0) # 0, u1 # 0 and there is a critical point u2 # 0 with either Cq-1('F,u2) # 0 or Cq+1(4,u2) # 0. When q ? 2, u2 # u1 since Cq-1('F,u1) = Cq+I('F,u1) = 0. 0

3.11. Cohomological Local Splitting The notion of cohomological local splitting is useful for obtaining nontrivial critical groups at zero and hence nontrivial critical points via Proposition 3.28 (ii).

Definition 3.33. We say that 'F has a cohomological local splitting near zero in dimension q, 1 < q < co if there are (i) a bounded symmetric subset M of W\ {0} that is radially homeomorphic to the unit sphere, and disjoint symmetric subsets A0

and B0 of M such that

0

i(Ao) = i(M\Bo) = q,

(3.32)

(ii) a homeomorphism h from IM onto a neighborhood U of zero containing no other critical points, such that h(0) = 0 and (3.33)

'FIA S 0 < 'FIB\{o}

where A = h(IAo) and B = h(IBo) u {0}.

Proposition 3.34. If 'F has a cohomological local splitting near zero in dimension q, then Cq('F, 0) # 0. PROOF. (3.33) gives the commutative diagram (Do n U

'F6 n U\{0}

1

1

A

h(Ao)

hl

h1

IAo

A0

U\B

h

ki

k1

h(M\Bo)

IM\(IBo u {0})

h

M\Bo

h1

M\Bo

where i, j, j, k, k, 1, and the unlabeled maps are inclusions. Passing to the induced diagram on reduced cohomology in dimension q - 1, j* # 0 since j* # 0 by (3.32) and Proposition 2.14 (iv) and the h* are isomorphisms. Since the radial projection onto M is a homotopy equivalence, k*, and hence also k*, is an isomorphism. So the top middle square gives 1* # 0. On the other hand, the top left square gives l*i* = 0 since IAo, and hence also A, is contractible. So i* is not onto. Therefore 8, and hence also Cq('F, 0), is

3. CRITICAL POINT THEORY

62

nontrivial in the exact sequence n U\ {0})

Hq

b Hq('k°nU,V nU\{0}) of the pair ('F° n U,'F° n U\ {0}).

O

Remark 3.35. Definition 3.33 is a variant of the notion of homological local linking introduced by Perera [96], which also yields a nontrivial critical group at zero.

3.12. Even Functionals and Multiplicity In this section we assume that 'F is even and use the cohomological index to obtain multiple critical points. Let F denote the class of symmetric subsets of W\ {0}, and for k < din N, let

Fk={MeF:i(M)>, k}

(3.34)

and

Ck = inf sup 'F(u).

(3.35)

ME.Fk uEM

Since Fk Fk+1, Ck < ck+l, and since S6(0) e Fk for any 6 > 0 by (i8) and sup 'F(8(0)) --+ 0 as S \ 0 by continuity, Ck < 0.

Proposition 3.36. Assume that 4 is even. (i) If -co < ck = ... = ck+m,_I = c < 0 and 0 satisfies (C), then i(K`) >, m. In particular, if -co < Ck < .. < ck+,,,_1 < 0 and 0 satisfies (C), for c = ck,... , ck+m._1, then each c is a critical value and 'F has m distinct pairs of associated critical points. (ii) If -co < Ck < 0 for all sufficiently large k and 4D satisfies (C) for all c < 0, then Ck / 0. First a lemma.

Lemma 3.37. If c < 0 and 4D is even and satisfies (C), then there is an e > 0 such that c - 6 < Ck < - < ck+m-I < c + E i(K`) > m. (3.36) PROOF. Since K° is compact by (C), there is a d > 0 such that i(Ns(K`)) = i(KO) (3.37) by (i4), and there are an e > 0 and an odd map i e C(W, W) such that i7(V+E\N6(KC))

(3.38)

c 'FC-E

by Lemma 3.7. Then (3.39)

i('`+E) <

i(Ns(K`)) < i('`-6) + i(K°)

3.13. PSEUDO-INDEX

63

by (i2), (i5), and (3.37). If c - e < ck, then 4)'-' 0 Fk and hence i($C_E) < k - 1,

and if c + E > ck+m_1, then there is an M e Fk+m-1 with M c $C+E and hence

i(4)C+8) > i(M) > k + m - 1 by (i2), so (3.36) follows from (3.39). PROOF OF PROPOSITION 3.36. (i) Taking c = ck =

= Ck+m-1 in

Lemma 3.37 gives i(KC) > in. Then taking in = 1 gives i(KCk) > 1 and hence KCk # 0 by (i1), so each ck is a critical value. Either they are all distinct and therefore have distinct pairs of critical points, or some c is repeated, which then has i(KC) > 2 and therefore infinitely many pairs of associated critical points by Proposition 2.14 (i). (ii) If ck / c < 0, then taking e > 0 as in Lemma 3.37 and k so large that ck > c - e gives i(KC) = co since ck+m_1 C c for all in, contradicting the compactness of KC by Proposition 2.14 (ii).

Proposition 3.38. If A, BC e F with

i(A) > k+m-1,

i(BC)
for some k, m > 1 and -oo < a:= inf $(B) < sup 4)(A) =: b < 0,

and 4) is even and satisfies (C)C for all c e [a, b], then a < ck < Ck+m-1 < b and hence there are m distinct pairs of critical points in 'a.

PROOF. Each M e Fk satisfies i(M) > i(BC) and hence intersects B by (i2), and > a on B, so Ck > a. Since A E= Fk+m_I and 4) C b on A, ck+m-1 < b. So the conclusion follows from Proposition 3.36.

Remark 3.39. Since A E Fk+m_I c Fk intersects B, inf 4)(B) < sup 4)(A).

3.13. Pseudo-Index The notion of a pseudo-index introduced by Benci [16] is useful for, among other things, obtaining multiple critical points of an even functional at positive levels.

Definition 3.40. Let F be as in the last section, M e F be closed, 0 C a < b < +co, and denote by r the group of odd homeomorphisms of W that are the identity outside D-1(a, b). Then the pseudo-index of M E F related to i, M, and r is defined by i*(M) = min i(ry(M) n M). 7Er

The following proposition lists some properties of the pseudo-index

i*:F--*Nu{0,oo}. Proposition 3.41. Let A, B e F.

3. CRITICAL POINT THEORY

64

(i) If A c B, then i*(A) < i*(B).

(ii) If rl e F, then i*(rl(A)) = i*(A). (iii) If A and B are closed, then

i*(A u B) < i*(A) + i(B). PROOF. (i) For each -y e r,

i(7(A) n M) < i(7(B) n M) by (i2). (ii) We have *(1!(A)) = min i(777(A) n M) = i*(A) 'yCr

since {7n : 7 E P} = F.

(iii) For each 7 e r, i(7(A u B) n M) = i((7(A) n M) u (7(B) n M))

6(7(A) n M) + i(7(B) n M) by (i5) and

i(7(B) n Al) < i(7(B)) = 2(B) by (i2).

For k < i(M) in N, let

Fk={MeF:i*(M)> k} and

ck = inf sup <(u). MEFF ueM

Since Fk D Fk+1,ck < ck+1

Proposition 3.42. Assume that < is even. (i) If a < ck = = ck+m-1 = c < 6 and 1) satisfies (C),, then

i(K`) , m. In particular, if a < ck <

< ck++n-1 < b and 1) satisfies (C), for e = ck,... , ck+, _1, then each c is a critical value and 4) has irn distinct pairs of associated critical points.

(ii) If a = 0, b = +co, 0 < ck < +oo for all sufficiently large k, and < satisfies (C), for all c > 0, then ck / +oo. As in the proof of Proposition 3.36, it suffices to prove the following lemma.

Lemma 3.43. If c > 0 and cF is even and satisfies (C), then there is an e > 0 such that c - E < C* -< " . < Ck+m-1 < c + e i(K`) >,m.

3.14. FUNCTIONALS ON FINSLER MANIFOLDS

65

PROOF. As in the proof of Lemma 3.37, there are b, e > 0 and t e IT such that (3.37) and (3.38) hold. So i*((I +e) < i*(4)c+e\No(Kc)) + i(N5(K`)) c i*(4)C-E) + i(K`)

by Proposition 3.41, and the conclusion follows as before.

Proposition 3.44. If M E F is bounded and radially homeomorphic to the unit sphere, U = IM; A, B E F with A c UC compact and B c M,

i(A)>k+m-1, for some k, m

i(M\B) Sk-1

1, and sup 4b(A) < a < inf 4)(B) C sup 4P(IA) < b,

and 4 is even and satisfies (C), for all c e (a, b), then a < ck 5 .. 5 ck+m-1 < b and hence there are m distinct pairs of critical points in l-1(a, b). PROOF. We apply Proposition 3.42. Each M e Fk satisfies

i(M n M) > i*(M) > i(M\B) and hence intersects B by (i2), and inf '(B) > a, so ck > a. If ry e IT, noting that -y1 A = idA since (b < a on A and applying Corollary 2.13 with Vi = ryIIA gives

i('y(IA)nM)>, i(A)> k+m-1. So i*(IA) > k + m - 1 and hence IA e Fti+m.-1 Since sup 4 )(IA) < b, then Ck+M-1 < b.

Remark 3.45. Since IA e Fk+ _1 c .F intersects B, inf'F(B) C sup 41, (1A).

3.14. Functionals on Finsler Manifolds A C1-manifold modeled on a Banach space V is a connected Hausdorff space M together with a collection {(U,, ep1\)}aeA, called an atlas on M, of pairs (UT, coa), called charts, such that (i) {Ua},EA is an open covering of M, (ii) for each A e A, Wa : U,, cpa(U,,) c V is a homeomorphism, (iii) for each A,li, e A, p, o coal (pa(U, n Uµ) cpµ(U, n Uµ) is a C'-diffeomorphism. We refer to Lang [64] for the basic constructions on M such as the tangent bundle TM and the cotangent bundle T*M. A Finsler manifold is a regular C'-Banach manifold M together with a continuous function TM --* [0, co), called a Finsler structure on TM, such that

(i) for each u e M, the restriction TTM at it is a norm,

IIu of

.

to the tangent space

3. CRITICAL POINT THEORY

66

(ii) for each u e M and C > 1, there is a trivializing neighborhood U of u such that

1111us11 -11.'
(3.40)

llv

VveU.

llwllu = sup *v), w E TT*M ve uM IIvIIv=1

where TTM.

is the duality pairing between the cotangent space T,*M and

The Finsler structure also induces a metric 1

d(u,v) =

IIQ (t)IIo(t) dt

inf

cEC1(0,11,M)

0

o(0)=u, o(1)=v

on M, called the Finsler metric, that is consistent with the topology of M. We assume that M is complete with respect to d. If 4' e C'(M,R), then 4'(u) e T,*M and the terminology and notation at the beginning of this chapter'-and Definition 3.1 still apply. We refer to Corvellec, Degiovanni, and Marzocchi [34] for the proofs of

Lemma 3.46 (First Deformation Lemma). If c E 1W, eo, b > 0, and 4' satisfies (PS), then there are c > 0 and rl e C([O,1] x M, M) satisfying (i) d(77(t, u), u) < bt V(t, u) e [0,1] x M,

(ii) 77(t, ) is the identity outside 4'CE0 ' for all t e [0, '1,

(iii) 4r(77(t, u)) < 4'(u)

V(t, U) E [0, 1] x M, (iv) 77(1, 4iV+E\Nj(K')) c P°-E.

Lemma 3.47 (Second Deformation Lemma). If -oo < a < b < +oo and 4' has no critical values in [a, b] and satisfies (PS), for all c e [a, b] n 1W, then 4'a is a deformation retract of 4'6.

Much of the theory of the previous sections can now be adapted to this setting.

Proposition 3.48. If c := inf 4'(M) is finite and 4' satisfies (PS),, then c is a critical value of 4'. PROOF. If not, let e > 0 and 77 e C([O, 1] x M, M) be given by Lemma 3.46 and take u e M with 4'(u) < c + E. Then 4(77(1, u)) < c - e by (iv), a contradiction.

Let A and B be disjoint nonempty subsets of M and

r = {y c C(CA, M) : 71 A = idA}. We say that A homotopically links B if

y(CA)nB#0 VyeP.

3.14. FUNCTIONALS ON FINSLER MANIFOLDS

67

Proposition 3.49. If M is a free 7G2-space and A and B are disjoint nonempty symmetric subsets of M such that

i(A) = i(M\B) < oo,

(3.41)

then A homotopically links B.

PROOF. If not, there is a -t e C(CA, M\B) with 71A = id A. Then

SA -. M\B, (u, t)

t E [0, 1]

ry(u, t),

7(-u, -t), t e [-1, 0)

is an odd map and hence

i(M\B) > i(SA) = i(A) + 1 by (i2) and (i6), contradicting (3.41).

Proposition 3.50. If A hornotopically links B, c:= inf

(3.42)

sup

4'(u)

ryEF uey(CA)

is finite, sup D(A) < inf'P(B), and 4D satisfies (PS)0, then c > inf'F(B) is a critical value of 4).

PROOF. If not, take eo < c - sup'P(A), let s and 11 be given by Lemma 3.46, and take rye P with sup'P(y(CA)) < c+s. Then 77(1, ) is the identity

on A by (ii), so 77(1, ) o y e F, but sup'P(r7(1, y(CA))) < c - s by (iv), contradicting (3.42). Now suppose that M is a free 7G2-space and 41, is even. The proof of the following symmetric deformation lemma can be found in Corvellec [32].

Lemma 3.51. If c e R, S > 0, and (D is even and satisfies (PS)0, then there are E > 0 and 77 e C([0,1] x M, M), with 77(t, ) odd for all t c [0, 1], satisfying (i), (iii), and (iv) of Lemma 3.46.

Let F denote the class of invariant subsets of M and for k < dim M in N, let Fk and ck be defined by (3.34) and (3.35), respectively. As in the proof of Lemma 3.37, it then follows from Lemma 3.51 that if c e R and 0 such that

i(K`)>, m. This in turn gives

Proposition 3.52. Assume that 9D is even. (i) If -CO < Ck = "' = Ck+m-1 = c < +co and lb satisfies (PS)c, then i(K`) in. In particular, if -co < ck S - - S ck+m_1 < +co and 'P satisfies (PS)0 for c = Ck, , Ck+m-1, then each c is a critical -

value and 4D has m distinct pairs of associated critical points.

(ii) If -oo < ck < +co for all sufficiently large k and $ satisfies (PS), then ck / +oo.

3. CRITICAL POINT THEORY

68

The following proposition gives the indices of the sublevel sets of 4'.

Proposition 3.53. Assume that 4' is even. (i) lick is finite and 4' satisfies (PS)0k, then i(M\cFCk) < k c i((D°k).

(3.43)

(ii) lick < ek+l are finite and 4' satisfies (PS)0 for e = ck, Ck+l, then

i(-`k) = i(M\4'a) = i(4'a) = i(M\4'ck+1) = k

Va E (Ck, Ck+1)

PROOF. (i) By Proposition 2.14 (iii), M\4'ek has a compact subset C e

F with i(C) = i(M\4'k). Then max4'(C) < ck since 4' is continuous, so C Tk and hence i(C) < k, and the first inequality in (3.43) follows. By (i4), 4'0k has a closed neighborhood N e F with i(N) = i((P°k). Since Kck is a compact subset of 4'Ck,

d := 2 dist(K°k, M\N) > 0. By Lemma 3.51, there are an'e > 0 and an odd map 71 E C(M, M) satisfying

d(rl(u),u) < b Vu e M,

E\Na(K°k)) C DCk-E

and hence

C DEk-E u N26(K`k) c N.

Taking an M e Tk with M c 4'`k+` then gives i(N) > i(4'`k+E) >, i(M) > k by (i2), so the second inequality in (3.43) also follows. (ii) By (3.43) and (i2),

k <, i(4>`k) S i(M\4'a) c i(4'a)

< k + 1.

Often M is of the form

M= In e W: 1(u) =1) with 1 a regular value of I E C' (W, ]k). Then M is a Cl-Finsler manifold by the implicit function theorem, and M is complete by the continuity of I. Moreover,

TuM = {v e W : (I'(u), v) = 0} = ker I'(u). Proposition 3.54. If 4' is a Cl-functional defined in a neighborhood of M and 4 its restriction to M, then the norm of 4f'(u) e TuM is given by (U) 11U = min 11 W' (u) - µ

(U)11

3.14. FUNCTIONALS ON FINSLER MANIFOLDS

69

PROOF. We have

RIME =

(q'(u),v)

sup

by (3.40)

veker I' (u) Iv1I=1

= min IIV(u) - p P(u) jAeR

by Lemma 3.55 below.

Lemma 3.55. If L, M E W*, then II LIkerM I` = min IIL - ttMII'. PROOF. For each p e II$,

II LlkerMII* = sup (L,v) S sup (L-pM,v) = IIL-hMII*. vekerM IvII=1

By the Hahn-Banach theorem, there is an L E W' such that L = L on ker M and

ILII'

- II LIkerM I'

Since ker (L - L) D ker M, L - Z = µM for some p e 1R, so II LIker M

IO'

= IIL - pMII `

.

CHAPTER 4

p-Linear Eigenvalue Problems In this chapter we study the p-linear eigenvalue problem (4.1)

Apu = ABpu

in W*, where A. satisfies (Al) - (A4), BV : W W* is (BI) (p - 1)-homogeneous and odd, (B2) strictly positive:

(By u, u) > 0

du # 0,

(B3) a compact potential operator, and A E R. We say that A is an eigenvalue of (4.1) if there is a u # 0 in W satisfying (4.1), called an eigenvector associated with A. Then an is also an eigenvector associated with A for any a # 0 by (A1) and (B1), and (Apu,u) (4.2)

A=

Bp u, u)

>0

by (A2) and (B2). The set o(Ap, Bp) of all eigenvalues is called the spectrum of the pair of operators (Ap , Bp ).

Example 4.1. In problem (1.9), the usual choice is

`Bpn V) = f

:

o

Op u = A Iulp-2 u in Q

U=0 i Then the first eigenvalue Al is positive, simple, and has an associated eigenon 852.

function Cpl that is positive in Il (see Lindqvist [68, 69]). Moreover, Al is iso-

lated in the spectrum o(-Ap), so the second eigenvalue A2 = inf o(-Ap) n (A1, oo) is well-defined. In the ODE case 71 = 1, where 1 is an interval, the spectrum consists of a sequence of simple eigenvalues Ak / oo, and the eigenfunction Wk associated with Ak has exactly k - 1 interior zeroes (see, e.g., Drabek [46]). In the semilinear PDE case n 2, p = 2 also, o(-a) consists of a sequence of eigenvalues Ak / oo. In the quasilinear PDE case re > 2, p :f 2, increasing and unbounded sequences of eigenvalues can be defined using various minimax schemes, but a complete list of the eigenvalues of -AP is still unavailable. 71

4. p.LINEAR EIGENVALUE PROBLEMS

72

Our setting also includes weighted eigenvalue problems as the next ex-

ample shows.

Example 4.2. In problem (1.9), taking (BP it, v) = JZ V(X)

jUlp-2 UV

with the weight V(x) > 0 a.e. and V e LS(Q) for some n

> -, PSn =1,

p>n

gives

-AP u = AV (x) lulp-2 a

u=0

in S2

onDQ.

This includes singular weights such as V(x) = lxl-9, 0 < q < min {p, n}.

4.1. Variational Setting Since (4.3)

(IP(u),u) = (Apu,u) =PIP(u),

zero is the only critical value of Ip and hence it follows from the implicit function theorem that

M := {u e W : II(u) = 1}

(4.4)

is a Cl-Finsler manifold. Moreover, M is complete, symmetric, and radially homeomorphic to the unit sphere since Ip is continuous, even, and p-homogeneous. By (1.6), (4.5)

l 1/p

inf

(Col

u s EM Ilull

ll IMP

(co/

By Proposition 1.2, the potential Jp of BP satisfying Jp(0) = 0 is given by

JP(u) = 1 (Bi,u,u) P

and is p-homogeneous and even. By (B2), Jp >, 0 and > 0 on W\ {0}. So the functional T(u) U C W\ {0}

4(u)'

is positive and its restriction

T=`PIm is CI. We will show that the critical values and the critical points of i are the eigenvalues and the eigenvectors of (4.1).

4.1. VARIATIONAL SETTING

73

Since

V(U)

_ _ )Z =

BP u,

.' (u) = Ap u,

it follows from Proposition 3.54 that the norm of V(u) e T,*M is given by III'(u)Ilu =mien

(4.7)

IIAAPU+'P(u)2BPUII*.

Lemma 4.3. Eigenvalues of (4.1) coincide with the critical values of ',

i.e., A is an eigenvalue if and only if there is a it e M such that W'(u) = 0 and W(u) = A.

PROOF. By (4.7), P(u) = 0 if and only if (4.8)

p e R. If (4.8) holds, applying it to u gives Y(u)2 (Ap

Y(u) c 0, - p(u)2 IP(u) -

u, u)

so (4.8) reduces to (4.1) with A = (u). Conversely, if A is an eigenvalue and it e M is an associated eigenvector, A = 1P(u) = JP(u)

(u)

by (4.2), so (4.1) implies (4.8) with p = -IY(u).

Example 4.4. In Example 4.1, M = 5 In E WD,P(Q) : 1 f I vulp ='I, w(u) _

P

st

IuiP,

P J11

Lemma 4.5.

JP(u) = 1

P st

l

lnIP

satisfies (PS).

PROOF. Let (uj) c M be a (PS), sequence, i.e., II'Y'(uj)11Ui '0.

'L(ug) -* c,

By (4.5), (uj) is bounded, so a renamed subsequence converges weakly to some u e W since W is reflexive, and BP uj converges to some L e W* for a further subsequence since BP is compact. Then (u.7) 'P

by Lemma 3.4, so c o 0.

P

= /BBuj,uj)

P

(L,u)

#0

4. p-LINEAR EIGENVALUE PROBLEMS

74

By (4.7),

µj Apuj +l (uj)2BPuj -.0

(4.9)

for some sequence (µi) c R. Applying (4.9) to uj gives µj + l (uj) 0, so µj -. -c # 0. Now applying (4.9) to uj - u gives (Ap uj, uj - u) - 0 since (BP uj, uj - u) 0 by Lemma 3.4, so uj -. it e M for a subsequence by (A4).

Lemma 4.5 implies that the set

Ea={uEM:I'(u)=0, lY(u)=A} of eigenvectors associated with A that lie on M is compact and that the spectrum o(Ap, BP) = {A e ]l8 : V(u) = 0, (u) = A for some u e M} is closed.

4.2. Minimax Eigenvalues We now construct an unbounded sequence of minimax eigenvalues for the eigenvalue problem (4.1). Although this can be done using the Krasnoselskii genus as usual, we prefer to use the cohomological index in order to obtain a nontrivial critical group and later construct linking sets. Let F denote the class of symmetric subsets of M, let

K= {keN:k(d}, and for k e K, let

Fk={MEF:i(M)>, k} and Ak = inf sup 11(u). MEFk uEM

Since Fk D Fk+1, Ak < Ak+l, and since the intersection of M with any k-dimensional subspace of W is a compact set in Fk by (is), Ak is finite. When d < oo we set Ad+1 =

OC)

for convenience.

Theorem 4.6. Assume (A1) - (A4) and (B1) - (B3). Then (Ak)kEK is a nondecreasing sequence of eigenvalues of (4.1).

(i) If Ak = "' = Ak+m-1 = A, then i(Ex) >, in. In particular, there are d distinct pairs of eigenvectors on M. (ii) The smallest eigenvalue, called the first eigenvalue, is IP(u) > 0. u#0 ,IJ(u)

Al = min qf(u) = min ueM

4.2. MINIMAX EIGENVALUES

(iii) We have i(.M\ (Yak) < k

75

i(1PAk). If AAk < A < Ak+1, then

i(M\t,) = 2(P) = 2(M\'Yak+1) = k. (iv) If d = co, then Ak / oo. PROOF. (i), (iv) Follow from Proposition 3.52. (ii) The first equality holds since T1 contains all antipodal pairs of points in M by Proposition 2.14 (i), and the second follows from homogeneity. (iii) Follows from Proposition 3.53.

The spectrum o(Ap, Bp) may possibly contain points other than those of the sequence (Ak), and the eigenvalues Ak may possibly be different from

the standard ones

ak =

inf

sup ql(u)

MET uEM 7±(M)->k

where

'Y+ (M) := sup {k > 1 : 3 an odd continuous map Sk-1

M}

and

-y-(M) := inf {k >, 1: 3 an odd continuous map M

Sk-1}

are the genus and the cogenus of M e .F, respectively.

Proposition 4.7. We have (z) 4i = A1,

(ii) Ak SAk2. PROOF. (i) Since both the genus and the cogenus of an antipodal pair of points in M is 1, al = min lF(u) = Al uEM

by Theorem 4.6 (ii). (ii) If M e T and there are odd continuous maps then

Sk+-1

-> M

Sk -1

k+
7+(M) C i(M) Cry (M).

Remarks 4.8. The sequence (Ak) is the more standard one. For the p-Laplacian, the sequence (A+) was used to construct homotopically linking sets by Drabek and Robinson [47]. The sequence (Ak) was introduced and

used to obtain a nontrivial critical group by Perera [98]. It has also been used to construct homologically linking sets by Cingolani and Degiovanni [30] and homotopically linking sets via the piercing property by Perera and Szulkin [105].

4. p-LINEAR EIGENVALUE PROBLEMS

76

4.3. Nontrivial Critical Groups The functional associated with the eigenvalue problem (4.1) is $a(u) = Ip(u) - A Jp(u),

(4.10)

u E W.

When A 0 o,(Ap, Bp), the origin is the only critical point of $a and taking U = W in (3.22) gives (4.11)

By homogeneity, $° (radially) contracts to {0} via

$a, (u, t) H (1 - t) u and $O\ {0} deformation retracts to $ n M via $A x [0, 1]

($a\{0}) x [0,1]

-

$,A\{0},

(u,t)

(1 -t)u+t7rM(u)

where (4.12)

7rM : W\ {0}

M,

u

it

Ip(u)Ilp

is the radial projection onto, M. Thus,

$anM-0

bgOZ2,

(4.13)

Cq($,\,0)

{m_1(nM),

$A n M # 0

by Proposition 2.4.

Theorem 4.9. Assume (AI) - (A4), (Bi) - (B3), and A 0 o(Ap, Bp ). (i) If A < AI, then C4($a,0) x 6gOZ2-

(ii) If A > Al, then

C9($a,0)

H9-I(AA)

Vq.

In particular, CO((Da, 0) = 0. (iii) If Ak < A < Ak+l, then Ck($a, 0) # 0.

PROOF. For n e M, (4.14)

-,

$a(u) = 1 - r

so $o n M = t'. Since $A = 0 if and only if A < aI by Theorem 4.6 (ii), (i) and (ii) follow from (4.13). Since i($a) = k when Ak < A < Ak+I by Theorem 4.6 (iii), then (iii) follows from Proposition 2.14 (iv).

Corollary 4.10. If A 0 o,(Ap, Bp), then Cq($a, 0) # 0 for some. q.

4.3. NONTRIVIAL CRITICAL GROUPS

77

Remark 4.11. Theorem 4.9 was proved by Perera [98], where the sequence of eigenvalues (Ak) was introduced precisely to obtain the nontrivial critical group in Corollary 4.10. We will use the notations introduced in this chapter throughout the rest of the text.

CHAPTER 5

Existence Theory In this chapter we study the global behavior of the functional 4? and obtain solutions of equation (1.1), together with some information about their critical groups, under various assumptions on the nonlinearity f at infinity. We assume that 0 has only a finite number of critical points. We classify (1.1) according to the growth of the potential F as (i) p-sublinear if F(tu) lim 5 0 Vu # 0, (5.1) t-. tP (ii) asymptotically p-linear if (5.2)

limX

F(1u)


(iii) p-superlinear if (5.3)

limes

F(tu) = oo Vu # 0.

5.1. p-Sublinear Case If we strengthen (5.1) to

F(u) = 0,

lim hull-*

IIuhID

then

'F(u) %

(CO

+ 0(1)) Ilur,

so 4l, is bounded from below and coercive, i.e., lim 4?(u) = oo. hull--.M

More generally, we have

Theorem 5.1. Assume (Al) - (A4), (Bi) - (B3), and (5.4)

F(u)
for some A < Al and C > 0. Then (i) $ is coercive and satisfies (PS), (ii) $ is bounded from below, CQ(4?,00) z dgOZ2i and 4? has a global minimizer u with CQ('F, u) x 8QOZ2. 79

5. EXISTENCE THEORY

80

PROOF. (i) Since Jp 3 0, replacing A with max {A, 0} if necessary, we may assume that A 0, so (5.5)

4'(u) > I/1,(u) - A Jp(u) - C

I

1 - 1)

-. oo as

jju1jP - C

by (5.4)

by Theorem 4.6 (ii) and (1.6)

- Co. Every (PS) sequence is bounded by coercivity and hence has a convergent hull

subsequence by Lemma 3.3. (ii) 1i > -C by (5.5), and then the rest follows from Corollary 3.19.

Example 5.2. In Example 4.4, (5.4) holds if

F(x, t) 5 A ltjP + C V(x, t) P

for some A < Al and C > 0.

5.2. Asymptotically p-Linear Case We strengthen (5.2) by assuming that

f = ABp - g

(5.6)

for some A > 0 and a compact potential operator g : W W* satisfying g(u) = o(llullp 1) as llull - oo.

(5.7)

Then F(u) = 'DA(u) + G(u) where Fa is given by (4.10) and (5.8)

G = A Jp - F

(5.9)

is the potential of g satisfying G(0) = 0. By Proposition 1.2 and (5.7), (5.10)

C(u) = I (g(tu), u) at = o(IIull') as dull -> oo. I

Example 5.3. In problem (1.9), assume that

f(x,t) = AItIP 't -g(x,t) for some g e Car(Q x R) satisfying

g(x, t) = o(ltl"-1) as tl

co; uniformly in x.

Then we have

-Ap u = A lulp-2 u - g(x, u)

{

u=0

in S2

on aU,

5.2. ASYMPTOTICALLY p-LINEAR CASE

81

and (5.6) and (5.7) hold with (g(u), v) =

f2 g(x, u) v,

G(u) =

f

G(x, u)

where

G(x,t) = f g(x, s) ds. 0

To verify the (PS) (resp. (C)) condition for D, it suffices to show that every (PS) (resp. (C)) sequence is bounded by Lemma 3.3, and the following lemma is useful for this purpose.

0 and pj :_

Lemma 5.4. If D'(uj)

Ik

II

-* co, then a subsequence of

uj := 'uj/pj converges to an eigenvector v associated with A.

PROOF. We have 11 ujll = 1 for all j, so a renamed subsequence converges

weakly to some v, since W is reflexive, and Bpvj converges in W* for a further subsequence since Bp is compact. Then

uj - v) _

+(4"(uj)-g(ui),wj-u)

A/Bpuj wj

y0

by Lemma 3.4 and (5.7), so vj -> u for a subsequence by (A4). Now passing

to the limit in Ap uj = A Bp uj +

V (uj) -Ig (uj ) Yj

gives Ap u = A Bp u, and u # 0 since Hull = 1.

Since (5.9) and (5.10) imply that (5.4) holds with a slightly larger A, here we assume that Ak < A < Ak+l for some k e ftC such thatAk < Ak+1. Let M and "P be as in Section 4.1 and let AO = yak and Bo = Wak+i Then

i(Ao) = i(M\Bo) = k by Theorem 4.6 (iii), so A = RAo cohomologically links B = 7r7 (Bo) u {0} in dimension k - 1 for any R > 0 by Proposition 3.26. By (5.8), (4.14), and (1.6),

1)

4'(Ru) < G(Ru) -

(5.11)

and

/

4'(u) > I 1 -

(5.12)

k+1 Lemma 5.5. If $ satisfies (C),

RP,

) Co Iullp + G(u), P

u e AO

u e B.

1

(5.13)

Rimes

- 1) RP] \k

f G(Ru) - 1 L

= -co, uniformly in u e AO,

5. EXISTENCE THEORY

82

and (5.14)

inf

r/

l

c0 IluIIp

1

I

k+1

L

P

+ G(u)J > -co,

then C°(4i,oo) = 0, Ck(,D,co) # 0, and (1.1) has a solution u with Ck(-P,u) # 0.

PROOF. -b is bounded from below on B by (5.12) and (5.14), and for any a < inf 'F(B) and sufficiently large R, (3.31) holds by (5.11) and (5.13), so the conclusion follows from Proposition 3.25.

Lemma 5.6. If A > Ak, then (5.13) holds, and if A < Ak+i, then (5.14) holds.

PROOF. The first assertion follows since llimo

G(RRu)

= 0, uniformly in it e A0

by (5.10) and (4.5), and the second follows similarly.

Nonresonance. We say that (1.1) is nonresonant if A 0 o-(Ap , By) in (5.6).

Theorem 5.7. Assume (A1) - (A4), (Bi) - (B3), and (5.6) and (5.7) with A E (Ak, Ak+l)\Q(Ap, Bp) Then -

(i) P satisfies (PS),

(ii) C°(p, cc) = 0,

oo) # 0, and (1.1) has a solution it with

0.

PROOF. (i) Since A 0 v(Ap, Bp ), every (PS) sequence is bounded by Lemma 5.4 and hence has a convergent subsequence by Lemma 3.3. (ii) Follows from Lemmas 5.6 and 5.5.

Resonance. We say that (1.1) is resonant if A e o,(Ap , Bp). Then we make an additional assumption on the non-p-homogeneous part of cF given by 1 (4i'(u), u) = G(u) - 1 (9(u), u) H(u) _ -P(u) - P

in order to ensure that D satisfies the (C) condition. Note that (H(uj)) is bounded for every (C) sequence (uj). Denoting by.NV the class of sequences

(uj) c W such that pj :_ INujIi -> oo and uj := uj/pj converges weakly to some u # 0, we assume one of (H±) H is bounded from below (resp. above) and every sequence (uj) e N has a subsequence such that

H(tuj)

±co Vt '> 1.

In particular, no (C) sequence can belong to N.

5.2. ASYMPTOTICALLY p-LINEAR CASE

83

Lemma 5.8. If (H±) holds, then G is bounded from below (resp. above) and every sequence (uj) e N has a subsequence such that ±ao.

G(uy)

(5.15)

PROOF. We have

p H(tu)

G(tu)

d

dt

=

tP

lim t-.m

tP+1

G(tu) = 0 tp

pH(tu) G(u) = J 1

dt.

f+1 dt = 1, inf H 5 G C sup H and (5.15) for the subsequence in (H+) follows from Fatou's lemma. We can now prove

Theorem 5.9. Assume (A1) - (A4), (BI) - (B3), and (5.6) and (5.7). Then (i) 4) satisfies (C) if (H+) or (H_) holds, (ii) D is bounded from below, C"(4), oo) z 6qo Z2, and 4) has a global minimizer u with C'](4), u)

6go7G2 if A = Al and (H+) holds,

(iii) Co(4), co) = 0, C'(4), co) # 0, and (1.1) has a solution u with Ck(4), u) # 0 in the following cases: (a) A E [Ak, Ak+1) and (H_) holds, (b) A E (Ak, Ak+1] and (H+) holds.

PROOF. (i) If a (C) sequence is unbounded, then Lemma 5.4 gives a subsequence that belongs to N, contradicting (H+). (ii) 4i = 4a, + G is bounded from below since 4ia, = I, - Al J, >, 0 by Theorem 4.6 (ii) and G is bounded from below by Lemma 5.8, and then the rest follows from Corollary 3.19. (iii) We apply Lemma 5.5. (a) Since A < Ak+1, (5.14) holds by Lemma (5.6), and since A >, Ak, it suffices to show that

lim G(Ru) = -co, uniformly in u e A0

R,.

oo such to verify (5.13). If not, there are sequences (u.,) c Ao and Rj that (G(Rj uj)) is bounded from below. Then pj := R7 11u111 -. oo by (4.5), a renamed subsequence of v,j := R. ujlpj = uj/ IIu, 11 converges weakly to some

5. EXISTENCE THEORY

84

u since W is reflexive, and BA converges to some L e W*.for a further subsequence since BP is compact. We have

(L, u) = lim BPVj, uj)

= lim .

by Lemma 3.4

P

'p(ui) 1v9 IIP

by (4.5)

by Theorem 4.6 (ii)

and hence u # 0, so (R uj) e N. But, then G(Rj uj) -* -oo for a subsequence by Lemma 5.8, a contradiction. (b) Since A > Ak, (5.13) holds by Lemma (5.6), and since A 5 Ak+I and G is bounded from below by Lemma 5.8, (5.14) holds as well.

Example 5.10. In Example 5.3,

H(u) =

Jfn

H(x,u)

where

H(x, t) = G(x, t) -

g(x, t) t

is the non-p-homogeneous part of G(x, t). pWe claim that (H±) holds if (5.16)

H(x, t) 3 (resp. <) C(x) a.e.

for some C e LI(Q) and oo. H(x, t) ±oo a.e. as By (5.16), H is bounded from below (resp. above). If (uj) e N, then for a subsequence, aj -> v a.e. and hence H(tuj) _> (resp. <-)

f

Ju(zl#0

H(x, tpj uj (x)) + J

C(x)

±oo

Vt

1

u(z1=0

by Fatou's lemma.

Remarks 5.11. The conditions on H(x, t) in Example 5.10 were given by Perera [97]. Different conditions for the existence of a solution, without the information on its nontrivial critical group, were given by Drabek and Robinson [47].

5.3. p-Superlinear Case In addition to (5.3), we assume that F is bounded from below and, in order to ensure that (P satisfies the (PS) condition, Hv(u)

F(u) - I (.f(u),u)

is bounded from above for some µ > p.

5.3. p-SUPERLINEAR CASE

85

For any u on the unit sphere S, dt (ID (tu))

= pip- IIi(u) - (f(tu),u) [p 4) (tu)

(5.17)

Pt

S

where

ao := inf

(4)(tu)

- (p - p) F(tu) + p Hµ (tu)]

- ao)

I [(µ - p) F - µHµ > -oc,

so all critical values of 4) are greater than or equal to no. Note that no < 0 since F(0) = Hµ(0) = 0. By (5.3), (5.18)

4)(tu) = tp (Iv(u)

- F(tu))

-oo as t

oc

for any u e S. The following lemma describes the structure of the sublevel sets at infinity.

Lemma 5.12. For each a < no, there is a CI-map Ta : S

(0, cQ) such

that {tu : u e S, t it Ta(n)}

PROOF. We have 4)(tu) 5 a for all sufficiently large t > 0 by (5.18), and

4)(tu) 5 a

dt (4)(tn)) < 0

by (5.17), so there is a unique Ta(u) > 0 such that

t < (resp. _, >) TT(u)

8 (tu) > (resp. _, <) a,

and the map T. is C' by the implicit function theorem. Then W\ {0}, which is ^ S, radially deformation retracts to 4a = {tu : u e S, t ? Ta(u)} via (W\ {0}) x [0, 11 -> W\ {0}

,

(1 - t) u + tTa(irs(u)) irs(u), (u, t))

in,

u e (W\

u e V.

We can now prove

Theorem 5.13. Assume (Al) - (A4), F is bounded from below and satisfies (5.3), and H is bounded from above for some p > p. Then (i) 4) satisfies (PS), (ii) Cq(4),00) ^ bgd Z2, (iii) 4) is bounded from above and has a global maximizeru with Cq(4), u) bgdZ2 if d < oo.

5. EXISTENCE THEORY

86

PROOF. (i) If (uj) is a (PS) sequence, then (1.6) gives 1

co

P

1

/ 1

iiu7 lip

E) Ip(u7)

(I

_ 4)(ui) -

- (4)'(u9),ui) + HM(ui)

c oGiuj 11) + 0(1),

so (uj) is bounded and hence has a convergent subsequence by Lemma 3.3. (ii) Since any a < ao is less than all critical values and 4) is unbounded from below by (5.18), Cq(4p,00) x HQ-1($a)

by Proposition 3.15 (ii)

z Hq-r(S)

by Lemma 5.12

8qd Z2

by Example 2.5.

(iii) Since S is compact, so is a = {tu : u e S, 0 < t < Ta(u)} and hence 4) is bounded there. Since 4 < a outside (ba, then 4) is bounded from above everywhere, and the rest follows from Propositions 3.17 and 3.18.

Example 5.14. In problem (1.9), we claim that all the hypotheses of Theorem 5.13 are satisfied if 0 < F(x, t) 5

(5.19)

1

lti >,T

f (x, t) t,

W

for some p > p and T > 0. By (1.10), f (x, t), and hence also F(x, t), is bounded on bounded t intervals. Integrating (5.19) gives

F(x, t) > c(x) Its" - C V(x, t) where c(x) = min F(x,±T)/Tµ > 0 and C > 0, so F((tu)

J

t"-p

f c(x) gulp -

C 1",

o0 as t - 00 Vu # 0

and F(u) >, -C 101 where JQJ is the volume of Q. Finally, since HM,(x,t)

F(x,t) - µ f(x,t)t

is S 0 for tj > T by (5.19) and bounded for Itl < T, HM(u) = L HM(x, u)

is bounded from above.

Remarks 5.15. In the semilinear case p = 2 of Example 5.14, condition (5.19) was introduced by Ambrosetti and Rabinowitz [7] and the critical groups at infinity were determined by Wang [132].

CHAPTER 6

Monotonicity and Uniqueness In this short chapter we give simple sufficient conditions for equation (1.1) to have at most one solution and show how they can be verified in applications.

Definition 6.1. An operator T : W

W* is monotone (resp. strictly

monotone) if

(T(u) - T(v), u - v) > (resp. >) 0 Vu # v. Theorem 6.2. If AP is strictly monotone and -f is monotone, then (1.1) has at most one solution.

PROOF. If both ul and u2 are solutions, then (Ap ui - Ar, u2, ui - 112) = (f (ui) - f (n'2), u1 - u2) ( 0 and hence ul = 162.

A.

The following lemma is useful for verifying the strict monotonicity of

Lemma 6.3. If (Apu,v) 5

IIvII

IIuIV'

Vu,v E W

and the equality holds if and only if au = /3v for some a, )3 >, 0, not both zero, then Ap is strictly monotone. PROOF. As in (1.8),

(Apu-Apv,u-v) >- (IIuIIp-1-IIvrrr-')(IIuII

- IIvII)

0.

If the leftmost term is zero, equality holds throughout and hence (Ap u, v) _ 1

IIuIIp

11v 11, (Apv,u) = Ilvllp

1

lull, and IIuII = 11v 11. Then an = /3v for some

a, 0 > 0, not both zero, so either a = v = 0, or a = /3 > 0 and hence

u=v#0.

Example 6.4. In problem (1.9), -f is clearly monotone if f (x, t) is nonincreasing in t, and now we show that the strict monotonicity of Ap follows from Lemma 6.3. By Schwarz and Holder inequalities,

(Apu,v) = f IVulp-2 Vu _ Vv 5 87

f IVul" lVvl - IIuIIp-1 IIvII

88

6. MONOTONICITY AND UNIQUENESS

Clearly, equality holds throughout if au = Qv for some a, (3 , 0, not both zero. Conversely, if (Ap u, v) = IIuiIP-1 iIvjj, equality holds in both inequalities. The equality in the Holder inequality gives

a IVul = Q IVvI a.e.

for some a, /3 ? 0, not both zero, and then the equality in the Schwarz inequality gives

a Du = /3 Vv a.e.,

so an = /iv.

CHAPTER 7

Nontrivial Solutions and Multiplicity In many applications f(0) = 0 and hence equation (1.1) has the trivial solution u = 0, and we are interested in finding others. Throughout this chapter we assume that this is the case and there is only a finite number of solutions.

7.1. Mountain Pass Solutions First we obtain nontrivial mountain pass solutions of (1.1) in the asymptotically p-linear and p-superlinear cases assuming that either F(u) < al Jp(u)

(7.1)

Vu E Bp(0)

for some p > 0, or

F(u) for some 05A
AJp(u) +o(IIulip) as u -. 0

Lemma 7.1. If (7.1) or (7.2) holds, then the origin is a local minimizer of

t and hence Cq(4), 0) .: 6q° Z2.

PROOF. Since Ip(u)/Jp(u) >, Al for all u # 0 by Theorem 4.6 (ii), if (7.1) holds, then 4) (u) >_ Ip(u) - Al Jp(u) > 0 = -D (0)

Vu E Bp (0),

and if (7.2) holds, then

II(u)

>,1 -

Al

+ o(1) as u -. 0

by (1.6). Proposition 3.18 gives the critical groups.

Theorem 7.2. Assume (Al) - (A4), (Bi) - (B3), (5.6) and (5.7), and (7.1) or (7.2). Then (1.1) has a mountain pass solution u # 0 with 415(u) > 0 in the following cases: (a) A c (Ak, Ak+l)\Q(Ap, Bp), (b) A E [Ak, Ak+l) and (H_) holds, (c) A E (Ak, Ak+l] and (H+) holds.

PROOF. Follows from Corollary 3.31 since C°(F, 0) x 7L2 by Lemma 7.1 and C°(,D, oo) = 0 and 4 satisfies (C) by Theorem 5.7 in case (a) and Theorem 5.9 in cases (b) and (c). 89

7. NONTRIVIAL SOLUTIONS AND MULTIPLICITY

90

Theorem 7.3. Assume (Ai) - (A4), (B1) - (B3), F is bounded from below and satisfies (5.3) and (7.1) or (7.2), and Hp, is bounded from above for some

p > p. Then (1.1) has a mountain pass solution ul # 0 with 4?(ul) > 0. If 2 < d < oo. then there is a second solution u2 # 0 that is a global maximizer of 4? with 4?(u2) > 0.

PROOF. Since Co((1,0) .: Z2 by Lemma 7.1 and C°(4t,co) = 0 and 4? satisfies (PS) by Theorem 5.13, (b has a mountain pass point ul # 0 with 4?(uI) > 0 by Corollary 3.31. When 2 5 d < co, there is a global maximizer u2 with CQ('F, u2) x Sgd7G2 by Theorem 5.13 (iii), 'D (U2) >, 4?(ui) > 0, and O U2 # uI since C1(4?,u2) # Cl(4),uI).

Example 7.4. When p > n in Example 4.4, (7.1) holds if

F(x,t) < Pl Itip,

(7.3)

ItI 5 6

for some 6 > 0 by the Sobolev imbedding Wo''(S2) - C(Si). When p < n, we strengthen (7.3) to

F(x4) c A ItID,

ti C 6

p

for some 0 < a < A1. Then

F(x,t) <, P Itlp+Cytr' V(x,t) for some r c (p,p*) and C > 0 by (1.10), so (7.2) holds by the imbedding I

W4

'r(0) - L' (52).

7.2. Solutions via a Cohomological Local Splitting Next we obtain nontrivial solutions of (1.1) in the p-sublinear and psuper-linear cases assuming that either

Ak Jp(u) 5 F(u) < Ak+l Jp(u)

(7.4)

Vu e Bp(0)\{0}

for some k e K such that Ak < Ak+l and p > 0, or (7.5)

A Jp(u) + o(IjuiI') < F(u) 5 A Jp(u) + o(Uiu(V') as it

0

for some Akc?CA
PROOF. We take U = rIM, h(u) = ru, A0 = 4fak, and Bo _ Yak+i with M and as in Section 4.1 and r > 0 sufficiently small. Theorem 4.6 (iii) gives (3.32) with q = k. If (7.4) holds, taking r C (co/p)VVP p, we have U c Bp(0) by (4.5) and hence 1

( 4lrM (u)) <

(u)

51

VueU\{0}, (7rM(a))

7.3. NONLINEARITIES THAT CROSS AN EIGENVALUE

91

which implies (3.33). If (7.5) holds, then (u)

+Q(1) as v

0

(7rM(u)) by (1.6), which also implies (3.33) when r is sufficiently small. Taking r `l`(im(M)) +O(1)

1

1

smaller if necessary, we may assume that U contains no other critical points. Proposition 3.34 gives the nontrivial critical group.

Theorem 7.6. Assume (Al) - (A4), (B1) - (B3), (5.4), and (7.4) or (7.5). Then (1.1) has a solution ul # 0. If k > 2, then there is a second solution

u2#0. PROOF. Follows from Corollary 3.32 since Ck(1), 0) # 0 by Lemma 7.5 and 4' is bounded from below and satisfies (PS) by Theorem 5.1.

Theorem 7.7. Assume (A1) - (A4), (B1) - (B3), F is bounded from below and satisfies (5.3) and (7.4) or (7.5) with k < d - 1, and Hµ is bounded from above for some p > p. Then (1.1) has a solution ul # 0 with either

(D(ul) < 0 and Ck-1(4',ul) # 0, or 4'(ul) > 0 and Ck+1(4 u.l) # 0. If k + 2 < d < oo, then there is a second solution u2 # 0 that is a global maximizer of 4' with 4(u2) > 0.

PROOF. Since Ck(4', 0) # 0 by Lemma 7.5 and Ck(4', CC) = 0 and 4' satisfies (PS) by Theorem 5.13, 4' has a critical point ul # 0 with either

,D(ul) < 0 and C'-1(4',ul) # 0, or 4'(u1) > 0 and Ck+I(4',ul) # 0 by Proposition 3.28 (ii). When k + 2 < d < oo, there is a global maximizer u2 with Cq(4',u2) bgd72 by Theorem 5.13 (iii), 4'(u2) > 4'(0) = 0 since Ck(4', 0) # 0 and hence the origin is not a global maximizer by Proposition 3.18, and u2 # u1 since Ck-1(4 U2) = Ck+1((pu2) = 0

Example 7.8. When p > n in Example 4.4, (7.4) holds if pk ItIP <

F(x,t) <

A k+1

t1P,

0 < Itj < 6

for some 6 > 0, and when p < n, (7.5) holds if A

P

It < F(x,t) <

A

P

NIP,

iti <- b

for some Ak < A < A < Ak+l.

7.3. Nonlinearities that Cross an Eigenvalue Now we obtain nontrivial solutions of (1.1) in the asymptotically p-linear case assuming that (7.6)

F(u) > Ak JP(u)

Vu e W

for some k e K such that Ak < Ak+l. Let Ao = 1YAk,

7-l = {h e C(CAo, M) : h(CAo) is closed in M, h1A0 = idAa},

7. NONTRIVIAL SOLUTIONS AND MULTIPLICITY

92

and sup

A = inf

'(u).

he4{ ueh(CA.)

Each h e 7-l extends to an odd map h e C(SAo,.M) since hIAo = idA. and Ao is symmetric, and sup (h(CAo)) = since W is even. By (i2), (i6), and Theorem 4.6 (iii), i(h(SAo)) > i(SA0) = i(Ao) + 1 = k + 1, so h(SAO) E Fk+l and hence sup (h(SAo)) > Ak+1 It follows that A > Ak+1

Theorem 7.9. Assume (A1) - (A4), (B1) - (B3), (5.6) and (5.7) with A 0 v(Ap , BP ), and (7.6). If A > A and either (7.7) F(u), 0, or (7.8) F(u) S A Jp(u) + o(IIulV') as u 0 for some A < Ak+1, then (1.1) has a solution u # 0 with D(u) > 0. PROOF. By (7.6), .D(tu) <' -

\

k

(u)

Il P < 0,

u e Ao, t > 0.

Since A > A, there is a h e R such that A' := supiI(h(CAo)) < A. By (5.8) and (4.14), <.P(Ru) '< -I

A -1-G((Ru)

and by (5.10) and (4.5), G(Ru) RP

0 as R

RP,

ueh(CA0),R>0,

oc, uniformly in u e h(CAo).

It follows that 4) 5 0 on A = R(IAo u h(CAo)) for all sufficiently large R.

Let B0 = Then 4 > 0 on B = rBo when r > 0 is sufficiently small. Indeed, if (7.7) holds, taking r = (co/p)1/P p, we have B c Bp(0) by (4.5) and hence

D(ru) >

1-

Ak}1

rP > 0 V ru e B,

'I`(u)

and if (7.8) holds, then 41, (ru) > I 1 -

A +o(1)) rP as r Ak+1

Since

i(Ao) = i(M\Bo) = k

0, u e Bo.

7.4. ODD NONLINEARITIES

93

by Theorem 4.6 (iii), A homotopically links B when R > r by Proposition 3.23, and since A $ o-(Ap, Br ), 4? satisfies (PS) by Theorem 5.7. Applying Proposition 3.21, c >, inf 4?(B) > 0 defined by (3.27) with g : A c W and X = IA is a critical value of 4?, and there is a corresponding critical point

onBwhenc=0. Example 7.10. In Example 4.4, (7.6) holds if F(x, t) >'

Ak

p

Itlp

d(x, t).

Remarks 7.11. Theorem 7.9 in the p-Laplacian case is due to Perera and Szulkin [105]. The semilinear case p = 2 is a well-known result of Amann and Zehnder [4].

7.4. Odd Nonlinearities Finally we assume that f is odd, so that 4? is even, and obtain multiple solutions of (1.1) using Propositions 3.38 and 3.44.

Theorem 7.12. Assume (Al) - (A4), (Bi) - (B3), (5.4), and f is odd. If F(u) 3 A Jp(u) + o(IIulI1) as u -+ 0 for some A > Am, then (1.1) has m distinct pairs of solutions at negative (7.9)

levels.

PROOF. We apply Proposition 3.38, taking A = r P- with r > 0 sufficiently small and B = W. We have i(A) >, m by Theorem 4.6 (iii) and i(B") = 0 by (i1). By (7.9) and (4.5),

4i(ru)5-C

-1+0(1))VP

as r-->0,ue

so sup 4?(A) < 0 when r is sufficiently small. By Theorem 5.1, 41, is bounded

from below, and hence inf 4i(B) > -co, and satisfies (PS). The conclusion follows.

When (1.1) is asymptotically p-linear we only consider the nonresonant case for the sake of simplicity.

Theorem 7.13. Assume (AI) - (A4), (Bi) - (B3), (5.6) and (5.7) with A 0 a(Ap Bp), and f is odd. ,

(i) If A < Ak and (7.10) F(u) ? A Jp(u) + o(jIuIIP) as u 0 for some A > Ak+.,,,_l, then (1.1) has m distinct pairs of solutions at negative levels. (ii) If A > Ak+m-I and (7.11) F(u) < AJp(u) + o(IIuIIp) as it - 0

for some A < Ak, then (1.1) has m distinct pairs of solutions at positive levels.

7. NONTRIVIAL SOLUTIONS AND MULTIPLICITY

94

PROOF. Since A 0 a(AP, Br ), 4 satisfies (PS) by Theorem 5.7.

(i) We apply Proposition 3.38, taking A = r l Ik+n.-, with r > 0 sufficiently small and B = 7r j (IWTk) u {0}. We have i(A) > k + m - 1 and

i(BC) = i(1rc (M\tAk)) = i(M\Pak) < k - 1 by Theorem 4.6 (iii). By (7.10) and (4.5),

a

4,(ru)<-

Ak+m-1

-1+0(1))x7 as r

0,ue

so supob (A) < 0 when r is sufficiently small. By (5.8), (4.14), (1.6), and (5.10), A

(u) >' L

Tk) p + 0(1)] IIuIIP

\1

as dull -. cc, u e B,

so inf 4)(B) > -oo. (ii) By Proposition 2.14 (iii), M\WA has a compact subset C cF with

i(C) =

We apply Proposition 3.44, taking A = RC and B = r 41ak c rM with r > 0 sufficiently small and R > r sufficiently large. We have

i(A) = i(C) = k+m-1 and i(rM\B) 5 k - 1 by Theorem 4.6 (iii). By (5.8) and (4.14),

CRu)

A

maxW(C)

-1-

G(R u)

)

RP,

u e C, R > 0,

and by (5.10) and (4.5),

G(Ru) Rv

-> 0 as R , cc, uniformly in u e C.

Since max (C) < A by the continuity of F-, it follows that sup 4 (A) < 0 when R is sufficiently large. By (7.11) and (4.5), iD(ru)

(1 - a +0(1) ) r7' as r - 0, u e Yak, k

so inf 4D(B) > 0 when r is sufficiently small.

0

Theorem 7.14. Assume (A1) - (A4), (Bi) - (B3), F is bounded from below and satisfies (5.3), HH is bounded from above for some p > p, and f is odd.

If F(u) < a JP(u) + oU uIl") as u -. 0

for some A < Ak, then (1.1) has d - k + 1 distinct pairs of solutions at positive levels. If d = w, then there is an unbounded sequence of positive critical levels.

7.4. ODD NONLINEARITIES

95

PROOF. We will show that $ has in distinct pairs of critical points in $-1(0, oo) for any m S d - k + 1 in N. We apply Proposition 3.44, noting that 4D satisfies (PS) by Theorem 5.13, and taking A to be the intersection of SR(O) with any (k+m-1)-dimensional subspace of W and B = r Yak c rM,

with r > 0 sufficiently small and R > (p/co)1/1'r sufficiently large. A is a compact symmetric set with index k + m - 1 by (i8) and i(rM\B) < k - 1 by Theorem 4.6 (iii). By (5.18), sup 4 (A) S 0 when R is sufficiently large, and as in the proof of Theorem 7.13 (ii), inf b(B) > 0 when r is sufficiently small. When d = oo, the sequence of critical levels in (0, oo) given by Proposition

3.44 is unbounded by Proposition 3.42 (ii).

Remark 7.15. Theorem 7.13 in the p-Laplacian case is due to Perera and Szulkin 11051.

CHAPTER 8

Jumping Nonlinearities and the Dancer-Fucik Spectrum Consider the problem

Apu=aB. u+bBB u in W*, where Ap satisfies (Al) - (A4), BBi : W W* are (Bl) (p - 1)-homogeneous, (8.1)

(BZ) nonnegative:

(BPu,u)>0 VUEW, (B3) compact potential operators,

(B4) Bp + Bp = Bp, and a, b e R. The set E(Ap , Bp , Bp) of all points (a, b) e R2 such that there is a it # 0 in W satisfying (8.1) is called the Dancer-Full spectrum of the triple of operators (Ap, Bp , Bp ). By Proposition 1.2, the potentials Jp of BP satisfying Jp (0) = 0 are given by

JP(u)=1(BFu,u) P

and are p-homogeneous. By (B2 ), Jp (u) > 0 for all u E W. Moreover, (8.2)

JP (u) + Jp (u) = JJ(u) > 0 Flu # 0

by (B4) and (B2), so either Jp (u) > 0 or Jy (u) > 0 when it # 0.

If

(a, b) e E(Ap , Bp , By) and it # 0 satisfies (8.1), applying it to it and using (8.2) gives

Ip(u) =a J4 (u) + b Jp (u) < max {a, b} Jp(u) and hence max {a, b} > Al by Theorem 4.6 (ii), so E(Ap, Bp , By) c {(a, b) C R2 : max {a, b} > Ai }.

By (B4), E(Ap, Bp , Bp) contains the set { (.F, A)

:

particular, the sequence of points ((Ak,Ak))kEK'

Example 8.1. In Example 4.1, the usual choices are (Bp U, V) =+Jn (u±)p-1v 97

A E a(Ap, Bp) }, in

8. JUMPING NONLINEARITIES AND THE DANCER-FUCIK SPECTRUM

98

where u± = max {± u, 0} are the positive and negative parts of u, respectively:

AP u = a (u+)p-I - b (u

)P-I

mQ

u=0 i The Dancer-Fucfk spectrum EI, was introduced in the semilinear case p = on On.

2 of this problem by Dancer [38, 39] and Fu, 1ik [51], who recognized its significance for the solvability of (1.9) when

f (x, t) = at'- - bt- + o(jtI) as

Itl

oo, uniformly in x.

In the ODE case n. = 1, Fhcffc showed that E2 consists of a sequence of hyperbolic like curves passing through the points (Ak, Ak), with one or two curves going through each point, and Drabek [46] has recently shown that Ep has this same general shape for all p in this case. In the PDE case u > 2, E2 consists locally of curves emanating from the points (Ak, Ak), in particular, contains two strictly decreasing curves, which may coincide, such that the points in the square (Ak_I, Ak+I) x (Ak-i, Ak+l) that are either below

the lower curve or above the upper curve are not in E2, while the points between them may or may not belong to E2 when they do not coincide (see Cac [22], Cuesta and Gossez [37], de Figueiredo and Gossez [41], Margulies and Margulies [77], and Schechter [119]). Since c°I solves

Op u = aI (u+)P-I - b (u-)P-I in 52 {

l

u=0

on3Q

for arbitrary b and -cpI solves Op u = a (u+)P

t

I

- aI (u )P-I

u=0

in 52

onaQ

for arbitrary a, Ep contains the two lines aI x R and R x AI. A first nontrivial curve in Ep passing through (A2, A2) and asymptotic to aI x 1R and 1R x aI at infinity was constructed using a mountain pass scheme by Cuesta, de Figueiredo, and Gossez [36]. More recently, unbounded sequences of decreasing curves of Ep, analogous to the lower and upper curves of Schechter in the semilinear case, have been constructed using various nlinimax schemes

by Cuesta [35], Micheletti and Pistoia [83], and Perera [101]. We close this introductory section with

Proposition 8.2. The spectrum E(Ap, By , BP) is closed. PROOF. Let the sequence ((aj, bj)) c E(Ap, B,+, By) converge to (a, b) e lR2 and let uj # 0 satisfy (8.3)

Ap uj = aj BP uj + bj B-uj.

8.1. VARIATIONAL SETTING

99

Using (Al) and (Bl) to replace uj with uj/ Iuj 11 if necessary, we may assume that jjujIj = 1, so a renamed subsequence converges weakly to some u since W is reflexive, and By uj converge in W* for further subsequences since BP are compact. Then (Ap uj, uj

- u) = aj (BP uj, uj - u) + bj (BD uj, uj - u) - 0

by Lemma 3.4, so uj

u for a subsequence by (A4). Now passing to the

limit in (8.3) shows that u satisfies (8.1), and u # 0 since hull = 1, so (a,b)eE(Ap,BP ,Bp ).

8.1. Variational Setting For each s > 0,

By := BP + s By is a (p-1)-homogeneous compact potential operator from W to W* by (Bl ) and (B3 ), and the p-homogeneous potential JP of BP satisfying 0 is given by Jp (u) =

P

(BP u, u) = JP (u) + s Jp (u).

Since either Jp (u) > 0 or Jp (u) > 0 when u # 0, JD(u) > 0 for all u # 0,

so the functional

T-(u) = Jp(u)' is positive and its restriction

u E W\{0}

to the manifold Nl defined by (4.4) is C1. We will show that the points of E(Ap , Bp , By) that are on the ray b = so, a >, 0 are of the form (c, se) with c a critical value of 1y. Since

V8(u) = -`FS(u)z BP u,

it follows from Proposition 3.54 that the norm of 'F (u) e (8.4)

is given by

l-s(u)Ilu = mm en

Lemma 8.3. The point (c, sc) e E(Ap, Bp , Bp) if and only if c is a critical value of '3. PROOF. By (8.4), YS(u) = 0 if and only if

µApu+ii,(u)2Byu = 0

(8.5)

for some p e R. If (8.5) holds, applying it to it gives s

a=

u

-W3(u)2 (BP u,u

3

'3(u)2 IPu) = -W3(u) < 0,

S. JUMPING NONLINEARITIES AND THE DANCER-FUCIK SPECTRUM

100

so (8.5) reduces to

Apu = cBy u

(8.6)

where c = ;P-, (u). Conversely, if it e M solves (8.6),

c=

IP(u) p(U)

so (8.6) implies (8.5) with p =

Example 8.4. In Example 8.1, J (u) = P

f P

(u+)p

`I' s (u)

f

P

= r

(u+)p + s (u-)p

Lemma 8.5. 's satisfies (PS). PROOF. Let (uj) c M be a (PS), sequence. By (4.5), (uj) is bounded, so a renamed subsequence converges weakly to some it e W since W is reflexive, and BP uj converges to some L e W* for a further subsequence since BP is compact. Then ' (

1P s u

)=

P

P

Bnu UJ

(L u)

#0

by Lemma 3.4, so c # 0. By (8.4), pj Ap uj + &(uj)2 By uj

(8.7)

0

for some sequence (pj) c R. Applying (8.7) to uj gives PPj + Ps(uj) - 0, so µj -. -c # 0. Now applying (8.7) to uj - it gives (Ap uj, uj - u) --* 0 since (By uj, uj u) 0 by Lemma 3.4, so uj -. it e M for a subsequence by

-

(A4).

8.2. A Family of Curves in the Spectrum We now construct an unbounded sequence of decreasing (continuous) curves in E(Ap , Bp , BP ).

Lemma 8.6. If sl < 82, then xpsi ! q/s21

sl `Ysl < s2 ` '32.

PROOF. We have

`I'si-`I's2=(S2-Sl)JP'sikPs230 and

81 T,, - 82 T12 = (S1 - 82) JJ Wsi'I`s2 5 0.

8.2- A FAMILY OF CURVES IN THE SPECTRUM

101

First we construct a curve Cl passing through the point (Al, Al). Let

c1(s) = EnAf s(u) and

C1 = {(cl(s),scl(s)) : s > 0}.

Theorem 8.7. Assume (Al) - (A4), (B1) - (B3), and (Bl) - (B4 ). Then Cl is a decreasing curve in E(Ap, Bp +, BP) passing through (Al, A1).

PROOF. cl(s) is a critical value of Ws by Proposition 3.48 and hence C1 c E(Ap, Bp +, By) by Lemma 8.3. If 81 C s2, then (8.8)

Ci(Sl) ! C1(82),

Si C1(81) 1< S2 C1(s2)

by Lemma 8.6, so Cl is decreasing. To see that it is continuous, note that (8.8) implies 82

C1(s2) < C1(s1) (

C1(82), 81

Sl C1(s1) < C1(82) < Cl(S1) 82

and hence c1(sl) -* cl(s2) as 81 / S2 and cl(s2) -* c1(s1) as 82 N sl. Since

Jp +JD = Jp by (8.2), `' is the functional >Y defined by (4.6), so cl(1) = Al by Theorem 4.6 (ii) and hence C1 passes through (A1, A1).

Now we construct a curve Ck in the spectrum for each k 3 2 in IC such that Ak > Ak-1. Let Ak = tp ak-i rk = {ry E C(CAk, M) : 7I Ak = id Ak}+

Ck(s) = inf

sup

'P,9(u),

-YEFk uE7(CAk)

and Ck = {(Ck(S), s Ck(S)): Ak-1/Ak < S < Ak/Ak-1}.

Theorem 8.8. Assume (Al) - (A4), (Bi) - (B3), and (B1) - (B4 ). Then Ck is a decreasing curve in E(Ap, BP , BP) and ck(1) 3 Ak. PROOF. Let Bk = 'Yak. Then

i(Ak) = i(M\Bk) = k - 1 by Theorem 4.6 (iii) and hence Ak homotopically links Bk by Proposition 3.49. Since

max is, 1}

min {.s, 1

when Ak_i/Ak < s < Ak/Ak-1,

sup I3(Ak) 5

Ak

mAk

< < inf q1s(Bk) in {s,l 1} max {s, 1}

It follows from Proposition 3.50 that Ck(S) >,

Ak

max{s,1}

8. JUMPING NONLINEARITIES AND THE DANCER-FUNK SPECTRUM

102

is a critical value of 'Ifs and hence Ck c E(AP, Bp ,BP ). By Lemma 8.6, S1 Ck(SI) 5 82 Ck(s2)

ek(81) % ek(s2),

Vs1 <1 S2,

and the monotonicity and continuity of Ck follow as in the proof of Theorem 8.7.

8.3. Homotopy Invariance of Critical Groups The functional associated with problem (8.1) is (8.9)

'D(a,b)(u) = IP(u) - JPa'b)(u),

uEW

JPa'b)

= a J +b JD is the potential of Bya'6) = a B' +b Bp satisfying JPa'b)(0) = 0, and the origin is the only critical point of'D(a,b) when (a, b) ¢ E(AP , BPI, Bp ). We will show that the critical groups C'1 (1(a,b), 0) are where

constant on path components of II82\E(AP , Bp +, By ).

Taking U = W in (3.22) gives C4(4)(n,b)10)

= H'($(a b), $(a b)\ {0}),

and for it # 0, (8.10)

where (4.12).

JPa'b)(7IM(u))) IP(u)

'D(a,b)(u)

JPa,b)

= j(a,b)I M

and lrM is the radial projection onto M given by

Lemma 8.9. If (a, b) 0 E(AP , BP , Bp ), then (IPab))1

16g072,

=0

Cq(D(a b), 0)

4q-1((JPab))1),

(JPab))1

0.

PROOF. By homogeneity, 1)0 b) contracts to {0} via 'Do

(a b) X

[0,11 -' $(a b),

(u, t) '-' (1 - t) u

and 4D(a b)\ {0} deformation retracts to (D(a b) n M via ('D(a b)\ {0}) X [0,11--. 4)(a b)\ {0}

,

(u, t) ,--. (1

- t) U + t lrM (u).

Since (ab) n M = (Jpa'b))1 by (8.10), the conclusion follows from Proposition 2.4.

It follows from Proposition 3.54 that the norm of (J(ab))'(u) E T A4 is given by (8.11)

(JPa b))/(u)Ilu µm

IpAPu-13( a'b)u

Lemma 8.10. The point (a, b) E E(AP , Bp , Bp) if and only if 1 is a critical Jpa,b). value of

8.3. HOMOTOPY INVARIANCE OF CRITICAL GROUPS

103

PROOF. By (8.11), (JP°'b))'(u) = 0 if and only if p AP u = BPa,b) u

(8.12)

for some µ e R. Applying (8.12) to u gives p = JPa'b)(u), so u e M is a critical point of JPa'b) with JPa'b)(u) = 1 if and only if (8.1) holds. The special case (aj, b5) _ (a, b) of the following lemma shows that JPa'b)

satisfies (PS), for all c # 0.

Lemma 8.11. If (aj, bj)

(a, b) in

JPai,bi(uj) -s c

0,

1[f2,

(uj) c M, and Dai,bi)A,¢j)Il

II(

u.J -. 0,

then ¢subsequence of (uj) converges to a critical point of JPa'b) with critical

value c.

PROOF. By (4.5), (uj) is bounded, so a renamed subsequence converges

weakly to some u e W since W is reflexive, and BP uj converge to some L± E W* for further subsequences since BP are compact. Then (8.13)

Bpa"b')

uj = aj Bpuj + bj BP uj -a L+ + b L-.

By (8.11), (-"b') p3 Ap uj - BP uj

(8.14)

0

for some sequence (µj) c R. Applying (8.14) to uj gives µj-JPai b')(uj) -* 0, 0 c # 0. Now applying (8.14) to uj - u gives (AP uj, uj - u) 0 by (8.13) and Lemma 3.4, so uj --' a for a since (Bp-"b') uj, uj - u) AP a subsequence by (A4). Then a e M since M is closed, and AP uj and L± = Bp u by the continuity of AP and B1 , respectively. Hence

so PLj

BPai'bi)

uj -

(8.15)

Bpa'b)

u by (8.13) and

cA2u-BPa'b)u=0

by (8.14). So u is a critical point of ya'b) by (8.11), and applying (8.15) to u gives JPa'b)(a) = c. We are now ready to prove

Theorem 8.12. Assume (Al) - (A4), (B1) - (B3), and (Bl) - (B4 ). If there is a path in II82\E(AP, By , BP) joining the points (a0, bo) and (al, b1), then C9(4'(ao,b0), 0) ,: Cq((p(a1 bl), 0)

Vq.

Since such a path is compact and C9(4)(a b), 0) is determined by the homotopy type of (JPb))1 by Lemma 8.9, it suffices to prove the following local version.

104

8. JUMPING NONLINEARITIES AND THE DANCER-FUCIK SPECTRUM

Theorem 8.13. If (a, b) e R2\E(Ap, Bp +, Bp ), then there is a 8 > 0 such that Pa'b ))1

(JPa'b))1

V(a', b') e B5(a, b).

Since the set R2\E(Ap, Bp , BP) is open by Proposition 8.2, for all sufficiently small 8 > 0, B5(a, b) c R2\E(Ap , BP , Bp) and hence 1 is a regular value of Pa'b) for all (a', b') a B5(a, b) by Lemma 8.10. Lemma 8.11 gives the following stronger version.

Lemma 8.14. If (a, b) e R2\E(Ap, Bp , Bp ), then there is a 6 > 0 such that JPab) has no critical values in [1, 1 + 36/a1] for all (a', b') e B5 (a, b).

PROOF. If not, there are sequences 6l \ 0, (o'l,bj') e Bo,(a,b), and (u1) c M such that (ab)(ul)

1 -4

5 1 + 38l/a1,

(Jra' b'1)'(u1) = 0.

Then a subsequence of (ul) converges to a critical point of Pa'b) with critical value 1 by Lemma 8.11 and hence (a, b) e E(Ap, Bp +, B;) by Lemma 8.10, a contradiction.

PROOF OF THEOREM 8.13. Let 6 > 0 be as in Lemma 8.14 and let (a', b') e B5(a, b). Then IJPa'b)(u)

- JPa'b)(a) < la' - al JJ (n) + Ib' - bI Jp (u) Vu e M

< 6 Jp(u) < 8/A1

by Theorem 4.6 (ii), so (fla'b')) (JP(" .b)), D

1+a/a, D (1Pa'b))1+2b/a,

.

(JPa'b ))1+3b/a,.

Since JPa'b) has no critical values in [1, 1 + 28/Al], there is a deformation retraction 77: (Jpa'b))1 x [0, 1] -' (fla'b))1

of (JPa'b))1 onto (JPa'6) )1+25/A, by Lemma 3.47 applied to -Jpab) Similarly,

there is a deformation retraction 7 (,7p(, b'))1+d/a, x [0, 1]

-' (Jla'b'))1+d a,

of (Jp° b))1+s/a, onto (JPa''b')1+3s/a,. The map (JPa'b))1 x [0, 1]

(u, t)

-' (JPa'b))1, 77(u,2t),

0 < t < 1/2

rj (,q(u, 1), 2t - 1),

1/2


8.4. PERTURBATIONS AND SOLVABILITY

105

is then a deformation retraction of (JJ°b))i onto (JP' 'b))1+3,Y/A,, and is also a deformation retract of (JJ°'b))i since JP°b) has (P¢'b')1+35/x,

no critical values in [1,1 + 3S/A1], so

(Jna',)1+35/a, (JP°,b))I. Since p a) is the functional 1a defined by (4.10), the following corollary is immediate from Theorem 8.12 and Corollary 4.10. (

Corollary 8.15. If (a, b) is in a path component of R2\E(Ap , BP , Bp ) containing a point of the form (A, A), then C9((I)(a b), 0) # 0 for some q.

Remark 8.16. Theorem 8.12 in the p-Laplacian case is due to Dancer and Perera [40].

8.4. Perturbations and Solvability Now we consider the solvability of (1.1) when

f =aBP +bBp

(8.16)

-g=BP(a'b)

-9

for some (a, b) e R2\E(Ap , Bp , BP) and a compact potential operator g

W W* satisfying (8.17)

1) as

g(u) = o(Ilull

hull - co.

Then 'D (u) = (D(a,b) (u) + G(u)

where (D(ab) is given by (8.9) and

G=aJP +bJP -F=J(ab) F is the potential of g satisfying G(0) = 0. By Proposition 1.2 and (8.17), (8.18)

G(u) = f (9(tu), u) dt = o(IlulIp) as dull -' co. 0l

Example 8.17. In problem (1.9), assume that f(x,t) =

a(t+)P-1

-

b(t

)P-1

- 9(x, t)

for some g e Car(Q x IR) satisfying

g(x, t) = o(jtjP-1) as

Itl

oo, uniformly in x.

Then we have

Op u = a (u* )P-` - b (u-)P- - g(x, u) in U

u=0

on a Q,

and (8.16) and (8.17) hold with

(9(u), v) = f 9(x, u) v,

G(u) = fn G(x, u)

8. JUMPING NONLINEARITIES AND THE DANCER-FUOIK SPECTRUM

106

where

G(x, t) =

f t g(x, s) ds. 0

The following lemma, together with Lemma 3.3, shows that both 4 and 'D(a,b) satisfy the (PS) condition when (a, b) 0 E(Ap, BP , BP ).

oo, then a Lemma 8.18. If (8.17) holds, cF'(uj) 0, and pj :_ jujII subsequence of uj := uj/pj converges to a nontrivial solution u of (8.1). PROOF. We have Ilii 11 = 1 for all j, so a renamed subsequence converges weakly to some u since W is reflexive, and BPa'b) uj converges in W* for a further subsequence since BPa'b) is compact. Then u'j_

(Apv'j,

u)BPa,b)J

7I'1),

i

by Lemma 3.4 and (8.17), so uj -> u for a subsequence by (A4). Now passing

to the limit in Apuj =

B(a,b)

aj +

'(uj) -g(uj)

3 gives Ap u =

BP(alb)

u, and u # 0 since IIuII = 1.

For R > 0, let

UR=RIM={ueW:Ip(u)<, RPI. By (1.6), UR is a bounded neighborhood of the origin with boundary OUR =

RM. Since (a, b) 0 E(Ap, Bp', By ), Lemma 8.18 implies that the critical set of 4 lies in UR if R is sufficiently large. The following lemma shows that outside U2R the lower-order term G can be deformed away without changing the critical set.

Lemma 8.19. If (a, b) 0 E(Ap, Bp +, BP ), then there are an R > 0 and a C1-functional PR such that (i) PR(u) _

'D(u),

ueUR n E U2c1 0

(ii) all critical points of both (P and 'DR are in UR, and hence the solutions of equation (1.1) coincide with the critical points of 4DR also by (i),

(iii) 4R satisfies (PS),

(iv) C9(DR, co) ^ C4((P(a,b), 0)

Vq.

PROOF. Since 4P(a,b) has no critical points on M,

6:=

V(a'b)(n)D U GM

*

>0

8.4. PERTURBATIONS AND SOLVABILITY

107

by Lemma 3.3, and then =6RP-1,

uEn.t II(a6)(Ru)II

R>0

by homogeneity. It follows that (8.19)

as R -.oc

inf

UGM

since sup Ig(RM)II* = o(RP-1) by (8.17). Take a smooth function (p [0, co) [0,1] such that o = 1 on [0, 1] and co = 0 on [2P, cc) and set 'PR(u) = 4)(a,b) (u) + w(IP(u)/RP) G(u).

Then (i) is clear. Since jj(co(Ip/RP))'(Ru)fl* = Icp'(IP(Ru)/RP)IIIAP(Ru)11*/RP

ueM

= O(R-1),

by (1.5) and supIG(RM)I = o(RP) by (8.18), (8.19) holds with 4) replaced

by 'FR also. So for sufficiently large R, inf l1-D'(u)ll* > 0, (8.20)

inf II''R(u)II* > 0 ueU`f

UGUn`

and hence (ii) follows. O

By (8.20), every (PS) sequence for 'FR has a subsequence in UR, which then is a (PS) sequence of 4) by (i) and hence has a convergent subsequence. Since 'F(a b) and G are bounded on bounded sets and V is bounded, 'FR

is also bounded on bounded sets. By (ii), the critical values of 'FR are bounded from below by inf 'FR(UR). Taking the a in (3.25) to be less than both inf 'FR(U2R) and inf 'F(a b) (U2R), say a', gives CQ((DR, 00) = IIQ(W, 4>R)

= H (W,

,p(a,b))

since 'R and (P(a b) lie outside U2R, where (DR = '(a,6) by (2). Since the origin is the only critical point Of 'F(a b) and a' < (a b) (0), FI4(W, '(a b))

G"1(D(a b), 0)

by Proposition 3.12.

The main result of this section is

Theorem 8.20. Assume (A1) - (A4), (B1) - (B3), (Bl) - (B4), and (8.16) and (8.17). If (a, b) is in a path component of R2\E(AP , By , By) containing a point of the form (A, A), then (1.1) has a solution. PROOF. It suffices to show that the functional 'FR in Lemma 8.19 has a critical point. Since C9(4)R, co)

C9(4, (a,b), 0) # 0

for some q by Corollary 8.15, this follows from Proposition 3.16.

CHAPTER 9

Indefinite Eigenvalue Problems In this chapter we drop the condition (B2) in the eigenvalue problem (4.1) and allow Jp to change sign. Then the eigenvalues may be positive or negative, but not zero since u = 0 is the only solution of Ap u = 0 by (A2).

9.1. Positive Eigenvalues First suppose

(B2) the set

M+:= {ueM:J4(n)>0} is nonempty. Since Jp is continuous and even, M+ is a symmetric open submanifold of M. As in Section 4.1, the positive eigenvalues and the associated eigenvectors on M+ are the critical values and the critical points of the positive and even Cl-functional

TIM+_

1

Since Jp is bounded on the bounded set M+ by Proposition 1.1, '+ is bounded from below by a positive constant. If (ui) c M+ is a (PS), sequence of W+, then c 0, and the argument in the proof of Lemma 4.5 shows that c > 0 and that a renamed subsequence

c, then Jp(u) = 1/c > 0 converges to some it e M. Since 1/Jp(uj) So ,p+ satisfies (PS). In particular, the set Ex of and hence it e eigenvectors of A that lie on M+ is compact and the positive spectrum M+.

o+(Ap , Bp) consisting of all positive eigenvalues is closed. We also have the following deformation lemma for ,y+ even though M+ is not complete.

Lemma 9.1. If c, b > 0, then there are e > 0 and 77 e C([0, 1] x .A4+, M+), with 77(t, ) odd for all t e [0,1], satisfying (i) d(a7(t, u), u) < bt V(t, u) e [0, 1] x M+, (ii) 4`+(77(t, u)) <, >Y+ (u) V(t, u) e [0, 1] x M+, (iii) 77(1, (iD+)c+e\N5(Kc)) e ( +)c-e

PROOF. We apply Lemma 3.51 to 4) = - JpjM at the level a = -1/c. Its critical set at this level is also K° by Proposition 3.54. An argument similar to that in the proof of Lemma 4.5 shows that 4) satisfies (PS)z. Thus, there 109

9. INDEFINITE EIGENVALUE PROBLEMS

110

are e > 0 and

e c([0, 1] x M, M), with (t, ) odd for all t e [0,1],

satisfying

u), u) < St V(t, u) e [0,1] x M,

(9.1) (9.2)

ID (

(t, u)) < ID (u)

(9.3)

V(t, U) E [0,1] x M,

X1(1,(V+i\N6(K`))

IfueM+, then JJ(i (t, u)) > JP(U) > 0 by (9.2), so r1(t, u) e M+. Let czF (9.4)

1+cE'71=17I[0,1]xM+

E

Then (i), (ii), and (iii) follow from (9.1), (9.4), and (9.3), respectively.

Let d+ = i(M+). Then 1 < d+ < d by (BZ ), (i1), (i2), and (i8). Let

K+={kEN:k
(9.5)

and for k e K+, let

Fk =IM GFk:McM+}

and

(9.6)

ak = inf

sup lY+(u).

MET, UEM

Since Fk D Fk+l, A < Ak+v and since there is a compact set in F,+, by Proposition 2.14 (iii), ak is finite. When d+ < co we set (9.7)

A+ d++] = +co.

The next theorem now follows as in Section 4.2 (Proposition 3.52 applies here since we have Lemma 9.1). Theorem 9.2. Assume (A1) - (A4), (B1), (BZ ), and (B3). Then (ak )kEK+ is a nondecreasing sequence of positive eigenvalues of (4.1). (2) If At = ... = '+m-1 = A, then i(Ea) >, m. In particular, there are d+ distinct pairs of eigenvectors on M+. (ii) The smallest positive eigenvalue is IP(u) al = LEM+ min +(u) = Jmin (u1>0 Jp(u)

> 0.

(iii) We have i(M+\(lY+)ak) < k < i((41+)ak ). If ak < A < ak+1, then

i(('P+)'k) = i(M+\('I'+)A) =

2(M+\( i-)Ak+1)

= k.

(iv) If d+ = oo, then At / +co. Turning to the critical groups of the functional 4ia in (4.10), we have

9.2. NEGATIVE EIGENVALUES

111

Theorem 9.3. Assume (Al) - (A4), (Bi), (B2 ) , (B3), and A 0 a+(A, , Bp).

(i) If 0
(ii) If A > A

,

then Hq-1((,y+)A)

Vq.

In particular, C°(4Da, 0) = 0. (iii) If AL < A < ak+1, then Ck(.I)A, 0)

# 0.

PROOF. For A >, 0 and it E M,

u*M+

31, ea(u)

A

so b'nA4 = (,Y+)A. Since

0 if and only if A < A by Theorem 9.2 (ii), (i) and (ii) follow from (4.13). Since i((1Y+)a) = k when ak < A < Ak+1 by Theorem 9.2 (iii), then (iii) follows from Proposition 2.14 (iv). 9.2. Negative Eigenvalues

Now suppose

(B2) the set

M :=fu eM:Jp(u)<0} is nonempty.

Applying the preceding discussion to -B, we see that M- is also a symmetric open submanifold of M and that the negative eigenvalues and the associated eigenvectors of (4.1) are the critical values and the critical points of the negative and even C'-functional

AIM-gyp

M_ which is bounded from above by a negative constant and satisfies (PS). The

set EX of eigenvectors of A that lie on M- is compact and the negative spectrum a-(Ap , BI,) consisting of all negative eigenvalues is closed. The index d- of M- satisfies 1 < d- < d. Let (9.8)

K-={kEN:k
and for k e )C-, let

F5 ={MEFk:MCM } and (9.9)

Ak = sup inf 91-(u). McFk

VEM

9. INDEFINITE EIGENVALUE PROBLEMS

112

Then Ak is finite and Ak+1 c Ak A. When d- < co we set ad-+1 = -co.

(9.10)

We have

Theorem 9.4. Assume (A1) - (A4), (B1), (B2 ), and (B3). Then (ak)kExis a nonincreasing sequence of negative eigenvalues of (4.1). (i) If Ak = .. _ Ak+m-1 = A, then i(EA) > m. In particular, there are d- distinct pairs of eigenvectors on M-. (ii) The largest negative eigenvalue is

Al = max W-(u) = max

I

P u) < 0. .]p(u)<9 JP(u)

UEM-

(iii) We have i(A lAA (Y )ak) < k

_ ).

a(( )

i(( )ak) 2(JV` \(

)

If Ak+1 < A < Ak, then

t(JV` \(

) k+1) = k.

(iv) If d- = oo, then Ak \ -oo. Theorem 9.5. Assume (A1) - (A4), (B1), (B2 ), (B3), and A 0 o (Ap , Bp). (i) If Al < A < 0, then 6gO7G2.

(ii) If A < Al, then Cq(.pa,0)

Hq-1((I )a)

Vq.

In particular, CO((DA, 0) = 0.

(iii) If Ak+1 < A < Ak, then Ck('DA' 0)

0.

9.3. General Case Since 00a(Ap,Bp),

a+(dd"P,Bp)uo,(Ap,Bp)=a(Ap,Bp)When neither, one, or both of a±(Ap , Bp) are empty, combining Theorems 9.3 and 9.5 gives

Theorem 9.6. Assume (AI) - (A4), (B1), (B3), and A 0 a(Ap B,). ,

(i) If A
,

then Cq('Da,0) x 6gO7G2.

(ii) If A < A

(resp. > A'), then

Cq(4)a,0) ^ Hq-1(( )a) (resp. Hq In particular, CO(4)a, 0) = 0 if A < A

+q or A > All

9.4. CRITICAL GROUPS OF PERTURBED PROBLEMS

(iii) Ifak+1 < A < A

or A' < A < Ak

1,

113

then

Cti((Da,0) # 0.

In particular, Corollary 4.10 holds without (B2). We emphasize that Theorem 9.6 applies in all possible cases:
,

(d) Jp

0: -oo =Aj
Example 9.7. When V is allowed to change sign in Example 4.2, 1

J4(u)=-P

f

d+=

V(x)Iulp,

st

00,

Ii 1>0

0,

52±j = 0

where Q± = {x e it : V(x) < 0}. Our setting also includes problems where the eigenvalue appears in the boundary conditions as the next example shows. Example 9.8. In problem (1.16), taking

(Bp u, v) = f V (X)

Iulp-2 UV

asZ

with V e Ls(Oul) and

n-1

pin

p-1

s

p>n

=1, gives

in Il

-Ap u + a(x) Iulp-2 u = 0 IVulp-2

av =

A V(x)

Then

- f V(x) IuIp, P on 1

Jp(u) =

d+ =

Inlp-27t

on OQ.

, Irtl > o o,

Ir±I = o

where r± = {x e 511: V(x) < Of and Ir±I is the area of r±.

9.4. Critical Groups of Perturbed Problems Returning to the equation (1.1), suppose

F(u) = AJp(u) +o(IIulIf) as it -> 0. The following proposition can be combined with the results on the critical (9.11)

groups of (D at infinity in Chapter 5 to obtain nontrivial solutions.

9. INDEFINITE EIGENVALUE PROBLEMS

114

Proposition 9.9. Assume (A1) - (A4), (B1), (B3), (9.11), and the origin is an isolated critical point of 4D.

(i) If A
(ii) Ifak+1
PROOF. (i) Since Ip(u)/JP(u) > al (resp. < Aj) when JP(u) > 0 (resp. < 0) by Theorem 9.2 (ii) (resp. Theorem 9.4 (ii)),

+O(1) 7rM(u)EM>0 P U) (9-121

>

+ o(1), ¶M (U) (= M

1P(u)

,

A< 0

1

otherwise

1 + o(1),

as u -> 0 by (9.11), where 7rM is the radial projection onto M, so the origin is a local minimizer of 4' and the, conclusion follows from Proposition 3.18.

(ii) We show that 4i has a cohomological local splitting near zero in

dimension k, with U = rIM and h(u) = ru where r > 0 is sufficiently small, and apply Proposition 3.34.

Case 1: ak < A < ak+. Let A0 = ( +)Tk and Bo = (`I'+)ak+l u (M\M+). Theorem 9.2 (iii) gives (3.32) with q = k, and (3.33) follows for sufficiently small r since A

H-1

-1+0(1))

,

7rM(u) E

k

(9.13)

IP(u)

1 - + +o(1),

9t4(u) E (F+)ak+l

k+1

7rM(u) e M\M+

1 + 0(1),

as u ->0 by (9.11). Let Ao = (Y )Ak and Bo = (T-) k+1 U

Case 2: Ak+1 < A < Ak.

(M\M-). Theorem 9.4 (iii) gives (3.32) with q = k, and (3.33) follows for sufficiently small r since 1

A - 1+0(l) /f

\Ak

(9.14)

A

+o(1),

,

7rM (u) E ( 7rM(u) E (r

k+1

+o(1), as u -. 0 by (9.11).

7r-M(u) E M\M-

9.4. CRITICAL GROUPS OF PERTURBED PROBLEMS

115

Now suppose (9.15)

F(u) = A Jp(u) + o(IIuIIP) as

co.

IIuII

We have

Proposition 9.10. Assume (Al) - (A4), (B1), (B3), (9.15), and 4) satisfies (C).

(i) If Aj < A < A', then Cg (4), oo)

Sgo Z2.

(ii) If ak+1
Ck(4), cw) # 0.

PROOF. (i) (9.12) holds as IIull .-> oo also by (9.15) and (1.6), so 4) is bounded from below and the conclusion follows from Corollary 3.19. (ii) We apply Propositions 3.25 and 3.26 with q = k-1 and R sufficiently large.

v Case 1: ak < A < At+1. Let A0 = ('Y+)Ak and Bo = k+1 (M\M+)]. Theorem 9.2 (iii) gives (3.28). (9.13) holds as IIull , co also by (9.15) and (1.6), so 4) is bounded from below on B and (3.31) follows for any a < inf 4)(B) and sufficiently large R. and B0 = [(Y-)ak+ U Case 2: Ak 1 < A < A. Let Ao = k (M\M-)]. Theorem 9.4 (iii) gives (3.28). (9.14) holds as IIull , Co also by (9.15) and (1.6), so 4) is bounded from below on B and (3.31) follows for any a < inf 4)(B) and sufficiently large R.

Example 9.11. In Example 9.7, (9.11) (resp. (9.15)) holds if

f(x,t) = A V(x) Itjp 2t-g(x,t) for some g e Car(fl x R) with subcritical growth and primitive G satisfying G(x, t) = o(I tIP) as tj -> 0 (resp. oo), uniformly in x. Here

Opu=AV(x)IuIP-`u-g(x,u) inc u=0

on aci.

Example 9.12. In Example 9.8, (9.11) (resp. (9.15)) holds if 0 (resp. cc), uniformly in x. F(x, t) = o(jtl2) as tI Here

-Ap u + a(x) Iulp-2 u = f (x, u) IVUl"-28v

in S2

=\V(x)Iul2-2u

on 03Q.

CHAPTER 10

Anisotropic Systems In this closing chapter we consider systems of equations of the form (1.1) where each equation may have a different p. Let m e N and for i = 1; ... , m, let (Wi, be a real reflexive Banach space of dimension di, 1 c di c co, with the dual (W*, II and the duality pairing -)i. Then their product -

W=Wlx...xWm={u=(ul,...,um):uieWi} is also a reflexive Banach space with the norm lull _

Iluill? i=1

and has the dual

W*=Wi x...xW,* ={L=(Ll,...,Lm):LieW*}, with the pairing mt

(L, u) =

(Li, ui)i i=1

and the dual norm 1

(\

IILII* = l G, (IlLilli

)2) 2

a-1

We consider the system of equations

AP u = F'(u)

(10.1)

in W*, where p = (pl,...,pm) with each pie (1,oo),

Apu = (Ap,ul, ... , Ap. um) Api e C(Wi,Wi*) is

(Ail) (pi - 1)-homogeneous and odd, (Ai2) uniformly positive: 3 co > 0 such that (Ap;ui, ui)i

Co IIuiIIP'

(Ai3) a potential operator, 117

Vu i e Wi,

10. ANISOTROPIC SYSTEMS

118

AP satisfies (A4), and F E C1(W, Ilk), F(0) = 0, and F' = (FF.,... , Fu,,,)

W W* is compact. As in Chapter 1, the potential IP of AP satisfying IP(0) = 0 is given by (10.2)

IP(u) = LJ

Pi

(AP:ui,'ui)i,

F can be written as F(u)

1

J0

(Fu; (tu), ui)i dt,

and the variational functional associated with equation (10.1) is

41(u) = IP(u) - F(u) 1

M

_

1

(APiui,

[Pi

ui)i

- J0 (Fui(tu), ui)i dtI

,

u E W.

Although not homogeneous, IP is even and satisfies (10.3)

.

co

1 Iluilir'

IIuiIIP' -< IP(-) -< Co

2=1 Pt

Vu E W

T=1 Px

for some Co >, co. Writing (u1/p1,. .. , un/pm) = u/p, we also have (10.4)

(IP(u), u/P) = (AP u, u/P) = IP(u),

analogous to (4.3).

Proposition 10.1. If each Wi is uniformly convex and Pi-1Il Pi < villi , ( APiua, u i)i = ri Iu i i (A Pt ui,vi)i - ri Iluilli

i, Vu z,

i

E Wi

for some ri > 0, then AP satisfies (A4).

PROOF. If uj - u and (AP uj, u? - u) m

0, then

Pi-1

(u

0

ti=1

uilli)

\AP'uj", ut

/i-

\APiua, ui/ i

- (APiui, ui ) + (APiui, ui)i )

_ (APu7,u7 - u) - (APu,ui -u) -, 0, so, for each i,

luti II

luilli and hence acs

ui by uniform convexity.

Example 10.2. Consider the p-Laplacian system

JDP u = Vu F(x, u) (10.5)

u=0

in 52

on8Q

10.1. EIGENVALUE PROBLEMS

119

where p = (p1, ... , pm) with each pi E (1, oo), u = (ui, ... , um), Ap u = (Ap,ui,..., Apmu.m), and F e Cl(S2 x W') satisfies F(x, 0) e 0 and OF

(10.6)

fVLi


ujl'"-1

\7

+1

V(x, u) e S2 x 1R"'

I

1

for some ri7 e (1, 1 + pj* (pi - 1) /p*). Problem (1.21) is the special case Pi = ... = PM = P. Here Wi = W01, Pi (Q), Vvi,

IVuIp.-2 Vui.

(Apini, vi)

Ip(u) =

12

M 1

Z Pi

IV1liIPi, f2

and the operator F is still given by (1.22). Ap satisfies (A4) by Proposition 10.1. 10.1. Eigenvalue Problems

We recall that a (continuous) representation of a topological group G on a Banach space W is a homomorphism p : G GL(W), where GL(W) is the group of invertible bounded linear operators on W, such that the map

(a, u) -p(a) u

G x W -. W,

is continuous. Then G also has the dual representation

p*:G-pGL(W*),

(p*(a)L,u)_(L,p(a 1)u)

on the dual space W*. For s = (s1, ... , s ) with each si e (0, w), the multiplicative group G = IR\ {0} of all nonzero real numbers has the representation

aui,..., Ialsm-1 aum) ps(a)u = on W = Wl x . . x W, with the dual representation (Ialsi-i

ps (a) L =

(IaI-sl-i

a Ll, ... ,

al-sm-1 a Lm)

on W*=Wi x...xW,*. We write (1/pi, ... ,1/p,,,) = 1/p and (1 - 1/pi,... , 1- 1/p,,,) = 1- 1/p, and let

Ua = p1/p(a) u =

(Tall/pi-1

a ui, ... , aI1/Pm-1 a u,,,)

and

L" = pi-1/p(a) L = (Tall/pi-2 aL1, ..., Then for any (10.7)

IaIl/pm-2 a L,,,,).

e C' (W, R),

(4'(ua)) =

(1all/pi-1 a4

= lal I'(Ua)a

(ua),..., Ial1/Pm-1adam (via))

10. ANISOTROPIC SYSTEMS

120

and

d (10.8)

da {q(ua)) _

(ua),

10,11/Pt-2aui/pi>i

i=1

= lal-1

/V (U.), ua/p) _ V (u'a)

u/P)

We have (10.9)

(Ap ua)a = (lal-1/P1 a Aplul,... , lal-1/P- a Ap,nu, )a = Ap u

and +n

(10.10)

Ip(ua) = E 1 /a'/ P= aAp.ui,

lull/Pt-1 aui>

i=1 p:

i

= lal "(U),

so we make

Definition 10.3. Let j be the subspace of C1(W, R) consisting of all functionals J satisfying (10.11)

J(ua) = lalJ(u)

Vu G W, a E G.

Then we have

Proposition 10.4. If J e J, then (i) J(0) = 0, (ii) J is even,

(iii) J'(ua)o = J'(n) Vu e W, a e G, (iv) (J'(u), u/p) = J(u) Vu E W. 0 and note that ua (ii) Take a = -1 in (10.11) and note that u_1 = -u. PROOF. (i) Take the limit of (10.11) as a

0.

(iii) Differentiate (10.11) with respect to u using (10.7).

(iv) Differentiate (10.11) with respect to a at a = 1 using (10.8).

Now we consider the eigenvalue problem (10.12)

Ap An = A J'(n)

in W*, where J e J with J' compact. (10.9) and Proposition 10.4 (iii) imply that if u is an eigenvector associated with A, then so is ua for any a e G and hence the entire orbit

0(u)={ua:aEG} of u is in the solution set of (10.12). We set no = 0 for convenience, and

note that the map II8 x W

W, (a, u)

ua is continuous.

Example 10.5. In problem (10.5), taking (`9Pi ui, vi) i = Ti ci

lVni p`-2 Vu . Vvi,

IP(u)

Ti

-

1

pi

IVniIPi S2

10.1. EIGENVALUE PROBLEMS

121

and

J(u) = J V(x) Jul I" ...

IU,mI

S?

with ri c (1, pi), m

(10.13)

Ti

= 1,

and V E LC0(Q) gives

r -AP;ui = A V(x) Iull'1 ... Iuilr.-2 ... u,,,l'mui in Q, i = 1,...,m

u1=

=um=0

onOQ.

Ap satisfies (A4) by Proposition 10.1 again, and (10.13) implies (10.11).

By (10.4), zero is the only critical value of Ip and hence it follows from the implicit function theorem that

M:= {uEW:Ip(u)=1} is a Cl-Finsler manifold. Moreover, M is complete and symmetric since Ip is continuous and even. By (10.3),

0 < inf hull <, sup lull < co.

(10.14)

,EM

uEM

The following lemma implies that M is radially homeomorphic to the unit sphere S in W.

Lemma 10.6. There is a C1-map T : S -1 (0, co) such that

M={T(u)u:uES}. PROOF. We have Ip(0) = 0, and for u e S and t > 0, m tpi

IP(tu) i=1 Pi

m tpi

(APiui,ui)i > co

- lluiIIP' -' oo as t -> oc

a=1 Pi

by (10.2), (Ail), and (Ai2, and co

dt

(I,(tu)) = L

tpi-1

(Apiui,ui)i >

1=1

tpi-i IIuillP > 0, Y 4=1

t > 0.

So there is a unique T(u) > 0 such that

t < (resp. =, >) T(u)

.

Ip(tu) < (resp. _, >) 1

and the map T is C' by the implicit function theorem. As in Chapter 9, we first consider the case where the set

M+:= {u 6M:J(u)>0} is nonempty. Since J is continuous and even, M+ is a symmetric open submanifold of M. We will show that the positive eigenvalues and the

10. ANISOTROPIC SYSTEMS

122

associated eigenvectors on M+ are the critical values and the critical points

of the positive and even Cl-functional

J M+ Since J is bounded on the bounded set M+, ,p+ is bounded from below by a positive constant. Since

(') (u) =

)2 =

II(u) = Apu,

+(u)2 J'(u),

it follows from Proposition 3.54 that the norm of ( +)'(u) e T, *M is given by

I('I`+)'(u)Ilw = min IltApu+lY+(u)2 J'(u)II".

(10.15)

jLrR

Lemma 10.7. Positive eigenvalues of (10.12) coincide with the critical values of 4l+, i.e., A > 0 is an eigenvalue if and only if there is a u e M+ such

that (W+)'(u) = 0 and i+(u) = A. PROOF. By (10.15), (lY+)'(u) = 0 if and only if (10.16) for some ju a R. If (10.16) holds, applying it to u/p and using (10.4) and

Proposition 10.4 (iv) gives

J(u) _ II(u)

_ +(u)2 (J'(u), u/P) = (Ap u,u/P)

0,

so (10.16) reduces to (10.12) with A = +(u). Conversely, if A > 0 is an eigenvalue and u e M+ is an associated eigenvector, A _ (Apu,u/P) = IP(u) =

(J'(u),u/P)

J(u)

All

+(u),

so (10.12) implies (10.16) with it = -lI+(u).

Example 10.8. In Example 10.5, ,n

M={(u1i...,um)eWo,Pl(52 )x...xWo,Pm(52):E J Vuilp'=1 l

i_1 Pi

4' + (u)

=

1

J V(x) lulIrl ... In4rm Lemma 10.9. + satisfies (PS).

12

11

10.1. EIGENVALUE PROBLEMS

123

PROOF. Let (ui) c M+ be a (PS), sequence. Then c > 0. By (10.14), (ui) is bounded, so a renamed subsequence converges weakly to some u e W

since W is reflexive, and J'(u3) converges to some L e W* for a further subsequence since J' is compact. Since the sequence (uj/p) is also bounded and converges weakly to u/p, then

T+(ui) =

I

1

us/p)

(L,u/p) by Proposition 10.4 (iv) and Lemma 3.4, so c > 0. By (10.15), (10.17)

t 3 Ap u' + +(u))2 J'(uj) - 0

for some sequence (p3) c R. Applying (10.17) to ui/p and using (10.4) and Proposition 10.4 (iv) gives lay +W+(uj) -. 0, so µi - -c # 0. Now applying

0 since (J'(u-'),ud - u) -. 0 by (10.17) to uu - it gives (Ap ui,ui - u u e M for a subsequence by (A4). Since 1/J(ug) - c, Lemma 3.4, so uj 0 then J(u) = 1/c > 0 and hence u e M+. Lemma 10.9 implies that the set Ea of eigenvectors of A that lie on M+ is compact and the positive spectrum a+(Ap, J') consisting of all positive eigenvalues is closed. Applying Lemma 3.51 to

JIM at the level a= -1/c, and noting

that an argument similar to that in the proof of Lemma 10.9 shows that satisfies (PS)a, we see that Lemma 9.1 still holds. So, defining K+ and ak by (9.5), (9.6), and (9.7), we have the following theorem (the second equality in (10.18) follows from (10.10) and (10.11)).

Theorem 10.10. Assume (Ail) - (Ai3), (A4), and J e ,7 with J' compact. Then (ak )kEK+ is a nondecreasing sequence of positive eigenvalues of (10.12).

A, then i(EA) > in. In particular, there = are d+ distinct pairs of eigenvectors on (ii) The smallest positive eigenvalue is

(i) If At =

M+.

(10.18)

(iii) We have

A = umin

+(u) = min J(ua) > 0. + a a#0

k

i((I+)A) =

(iv) If d+ = co, then ak J' +co. We now consider the case where the set

M- := {u E M : J(u) < 0}

If At < A < ak+1, then

k ) = k.

10. ANISOTROPIC SYSTEMS

124

is nonempty. Applying the preceding discussion to -J as in Chapter 9, we see that M- is also a symmetric open submanifold of M and that the negative eigenvalues and the associated eigenvectors of (10.12) are the critical values and the critical points of the negative and even C1-functional ip- _

1

J M_

which is bounded from above by a negative constant and satisfies (PS). The set Ea of eigenvectors of A that lie on M- is compact and the negative spectrum a- (AI,, J') consisting of all negative eigenvalues is closed. Defining 1C- and ak by (9.8), (9.9), and (9.10), we have

Theorem 10.11. Assume (Ail) - (At3), (A4), and J e ,7 with J' compact. Then (ak)kEK- is a nonincreasing sequence of negative eigenvalues of (10.12).

(i) If Ak = ..' = Ak+m-i = A, then i(E),) >, m. In particular, there

are d- distinct pairs of eigenvectors on M-.

(ii) The largest negative cigenvalue is

Al = max Y (u) = max IP(ua) < 0. uEM-

uEM- J(ua) a#0

(iii) We have i(M \('I )ak) < k

)ak ). If Ak+l < A < Ak, then

i(('Y )ak) = i(M \('I` )A) = i(('I )a) = i(M \(y )ak+) = k. (iv) If d- = cc, then Ak \ -oo. Since u = 0 is the only solution of Ap u = 0 by (A%2), 0 0 o(Ap , J') and hence

a+(Ap , J') u a (Ap, J') = a(Ap, J'). When A 0 a(Ap, J'), the critical groups of the associated functional

Fx(u) = Ip(u) - A J(u),

ucW

at zero are given by (4.11).

By (10.10) and (10.11),

'Pa(ua) = lal 'Pa(u), so O(4') c 4)0. Hence 4150 contracts to {0} via (u, t)

4)°x [0,1] -* 4' ,

ul_t.

If u A 0, au := Ip(u) > 0 by (10.3) and the path component of O(u) containing u intersects M at u := ua-1 since Ip(u.) = 1 by (10.10). Hence

$a\ {0} deformation retracts to 415a n M via {0}) x [0, 11 -* $O\ {0}

,

(u, t) '-' ul-t+tau'

Thus, (4.13) still holds, and as in Chapter 9 we get

10.2. CRITICAL GROUPS OF PERTURBED SYSTEMS

125

Theorem 10.12. Assume (Ail) - (Ai3), (A4), J e 9 with J' compact, and A

a(A,,J'). (i) If \-
Gg('Da, 0) ` bqO Z2.

(ii) If A < Ai (resp. > A'), then

Gq(.a,0) : Hq-1((p )a) (reap. H° 1((1'+)A))

Vq.

In particular, CO (,DA, 0) = 0 if A < Ai or A > Al .

(iii) If A-

I

< A < Aj orAk < A < Ak+1, then Ck(4'A, 0) # 0.

In particular, Cq(4)a, 0) # 0 for some q.

10.2. Critical Groups of Perturbed Systems Returning to the system (10.1), suppose

F=AJ-G

(10.19)

with

G(ua) = o(a) as a \ 0, uniformly in u e M.

(10.20)

Proposition 10.13. Assume (Ail) - (Ati3), (A4), J e J with J' compact, (10.19) and (10.20) hold, and the origin is an isolated critical point of 4P. (i) If Ai < A < Ai , then the origin is a local minimizer of b and (10.21)

C" ((D, 0) z b,qO Z2.

(ii) If A-

I

< A < Ak or Ak < A < Ak+1, then Ck((p, 0) 7, 0.

PROOF. For r > 0,

hr(tu) = urt,

u e M, t E [0, 11

defines a homeomorphism from IM onto

Ur={ua:aeM,0
hr 1 (u) =

auru, u#0 to,

u=0

where au and u are as in the last section. In particular, (Jr is a neighborhood of zero. By (10.19), (10.20), (10.10), and (10.11), (10.23)

I'(ua) = a (1 - AJ(u) +o(1)) as a \ 0, uniformly in u e M.

10. ANISOTROPIC SYSTEMS

126

(i) Since J(u) < 1/al (resp. > 1/Aj) for u e M+ (resp. M-) by Theorem 10.10 (ii) (resp. Theorem 10.11 (ii)),

A Al (10.24)

ueM+,A>0

+0(1)

4)(ua) % a( I - a +o(l) )

u e M-, A < 0

1

a (1 + o(1)),

otherwise

by (10.23), and (10.21) follows from Proposition 3.18.

(ii) We show that 4) has a cohomological local splitting near zero in dimension k, with U = U, and h = hr where r > 0 is sufficiently small, and apply Proposition 3.34. Note that

A=h,.(IAo)={ua:ueAo,05acr}, B=h,.(IBo)u{0}={ua:ueBo,0
u

(M\M+). Theorem 10.10 (iii)-gives (3.32) with q = k, and (3.33) follows for sufficiently small r since

-a A+ k

(10.25)

q(ua)

> a I\1

-

- 1 +o(1))

+

,

+ 0(1) /f

uE

wE

\k+l

k+1

uEM\M+

>a(1+O(1)), by (10.23). Case 2:

,+)+ (

Ak+1 < A < Ak. Let Ao = (lY-)ak and Bo = (T-) k+u

(M\M-). Theorem 10.11 (iii) gives (3.32) with q = k, and (3.33) follows for sufficiently small r since

-1+O(1) k

(10.26)

D(ua)

>a (1-

\

I\

/

+ o(1) I

\k+l

>a(1+o(1)),

,

ie k

,

uE(

k+1

J/

ueM\M

by (10.23).

Now suppose (10.19) holds with (10.27)

G(ua) = o(a) as a ' cc, uniformly in it c M.

Proposition 10.14. Assume (Ail) - (Ai3), (A4), J G J with X compact, (10.19) and (10.27) hold, and 0 satisfies (C).

10.2. CRITICAL GROUPS OF PERTURBED SYSTEMS

127

(i) If Aj < A < A', then 4) is bounded from below and (10.28)

01 (4,00) x 690ZG2.

(ii) If Ak+1
= 0,

1,

then

Ck(4),oo) # 0.

PROOF. Identifying W with {tu : u e M, t e [0, oo)}, hl(tu) = ut

defines a homeomorphism of W (hi 1 is given by (10.22) with r = 1). In particular,

W={ua:ueM,a>0}. By (10.19), (10.27), (10.10), and (10.11),

$(u0) = a (1 - A J(u) + o(1)) as a oo, uniformly in it e M. (i) (10.24) holds as a / oo also by (10.29), and (10.28) follows from

(10.29)

Corollary 3.19.

(ii) We apply Propositions 3.25 and 3.27 with q = k - 1, h = hl, and R sufficiently large. Note that A = h1(RAo) _ {uR : u e Ao

B=h1(7rj (Bo)u{0})={u0:ueBo, a>0}u{0}. Case 1: ak < A < ak+1 Let Ao = (IIJ+)a,+ , and Bo = [(1Y+)a+ u k+1 (M\M+)]. Theorem 10.10 (iii) gives (3.28). (10.25) holds as a / oo also by (10.29), so D is bounded from below on B and (3.31) follows for any a < inf 4)(B) and sufficiently large R. [(lIJ-)Ak+1 U Case 2: ak+1 < A < ak . Let A0 and B0 = k (M\M-)]. Theorem 10.11 (iii) gives (3.28). (10.26) holds as a / oo also by (10.29), so 4) is bounded from below on B and (3.31) follows for any a < inf $(B) and sufficiently large R. Example 10.15. In Example 10.5, setting

G(x,u) = AV(x) lull" ...

lu,lrm

- F(x,u)

gives

-Apiu'i = A V(x) In,

Iri

aG ... Iui Iri-2 ... IU,Irm ui - aui (x, u)

in Q,i=1,...,m u1=...=Um=0

on 8S2

and

G(u)

G(x, u).

10. ANISOTROPIC SYSTEMS

128

If

M

IG(x,u)I - C

'

Juils=

V(x,u) E Q x R'

i=1

for some si e (pi, p, ), then si/pi

G(ua)l C C

iui

llz,

i=1

and hence (10.20) holds. Similarly, (10.27) holds if

IG(x,u)l5C(M=1 E lullSi+l

V(z,u)eQxIl2"

for some si e (O,pi).

10.3. Classification of Systems The last section contains the critical group computations necessary to

extend the existence and multiplicity results of Chapters 5 and 7 to systems. We classify (10.1) according to the growth of F as

(i) p-sublinear if (10.30)

a/a lim

F(ua)

= 0, uniformly in u c M,

(ii) asymptotically p-linear if lim F(ua) = A J(u), uniformly in u e M a/M a for some J e 3 with J' compact and A # 0, (iii) p-superlinear if

(10.31)

(10.32)

lim F(ua) = 00,

a/'w

a

u e M.

Since (10.30) is the special case A = 0 of (10.31), we treat the p-sublinear and asymptotically p-linear cases together. Critical groups of 4) at infinity are given by Proposition 10.14. In order to ensure that satisfies the (PS) condition, we strengthen (10.31) by assuming (10.19) with (10.33)

lim G'(ua)a = 0, uniformly in u e M a/ce

in the notation of Section 10.1. Since

G(ua) - G(ull) =

f

d (G(ut)) A = f (G'(ut)t, u/p dt

for a > P > 0 by (10.8), then (10.27) holds and hence (10.31) follows from (10.11).

10.3. CLASSIFICATION OF SYSTEMS

129

Example 10.16. In Example 10.15, for a > 0,

l

(G (u«) « , v} _ i=1

1/p,-1

\IX

-

(' dG

1

_

Cui (ua), vi

i=1 a1-1/pi

J

aus

s2

(X ua) vi>

so if

aG dui

iuii si'-1 + 1)

C

for some sii e (1, 1 + pi (pi

-

V(x,u) e Q x Rm

1

1)/pi), then luilisi 3

IIG'(ua)all* 5 C i=1

1

1

a1-1/r; (.,-1)/P, + a1-1/p; j=1

and hence (10.33) holds.

Lemma 10.17. If (10.19) and (10.33) hold with A 0 a(Ap,J'), then 'F satisfies (PS). PROOF. It is easy to see that Lemma 3.3 still holds, so it suffices to show that every (PS) sequence (ui) is bounded, which is equivalent to the bound-

and note that

edness of ai := Ip(ui) by (10.3). If ai / co, set ui := u3

Ip(ui) = 1 by (10.10) and hence ui e M. We will show that a subsequence of (Ii) converges to an eigenvector u associated with A, contradicting the assumption that A a(Ap, J'). By (10.19),

AP n3 - A J' (ui) ='F' (u'') - G' (uu ), and hence (Ap ul)a' - A J'(u3)a' _ P'(uj)a' - G'(u3)a'.

Since u3 = vk

,

(Apu7)a' = ApU7,

J'(uj)ai = X(0'

G'(uj)aj -* 0

by (10.9), Proposition 10.4 (iii), and (10.33), respectively. For j so large that ai > 1, 1

ll* =

(a1/P_1

2

ui111

1D

*

II

)2

1

mnip;

.

Thus, (10.34)

AP W - A J' (W) -- 0.

By (10.14), (u3) is bounded, so a renamed subsequence converges weakly

to some it since W is reflexive, and J'(u3) converges in W* for a further subsequence since J' is compact. Applying (10.34) to ui - it gives (Ap W, W - u) ->0 since (J'(ui), ui - u) -. 0 by Lemma 3.4, so ui - it e M for a subsequence by (A4). Now passing to the limit in (10.34) gives Apii =AJ'(u).

10. ANISOTROPIC SYSTEMS

130

The following existence result is now immediate from Propositions 10.14 and 3.16.

Theorem 10.18. Assume (A 1) - (Ai3), (A4), J e j with J' compact, A 0 a(Ap, J'), and (10.19) and (10.33) hold. (i) If Aj < A < A', then 4) has a global minimizer u with Cq((D, u) 8q0 74.

(ii) If ak+1 < A < ak or ak < A < ak+l, then (10.1) has a solution u with Ck(4), u) # 0. Now suppose

F=A0Jo-Go

(10.35)

where Jo e J with JJ compact and

Go (u.) = o(a) as a \ 0, uniformly in it e M,

(10.36)

and let (Aok) be the sequences of positive and negative eigenvalues associated with Jo.

Theorem 10.19. Assume (Ail), - (Ai3), (A4), Jo, J e J With Jo', J' compact, A 0 o(Ap , J'), and (10.35), (10.36), (10.19), and (10.33) hold. (i) If Ao,k+l < Ao < aok or A k < Ao < aok+1, and AT < A < A , then (10.1) has a solution ul # 0. If k 3 2, then there is a second solution u2 # 0.

(ii) If AO-1 < Ao < A j, and ak+1 < A < ak or ak < A < ak+l, then (10.1) has a mountain pass solution it # 0 with 4)(u) > 0. PROOF. (i) Follows from Corollary 3.32 since Ck(4), 0) # 0 by Proposition 10.13 and 4) is bounded from below by Proposition 10.14.

(ii) Follows from Corollary 3.31 since Co(F, 0) x Z2 by Proposition 10.13 and C°(4), co) = 0 by Proposition 10.14.

O

In the p-superlinear case we also assume that F(u) and (F'(u), u/p - u/µ) are bounded from below and

Hp(u) := F(u) - (F'(u), u/W)

(10.37)

is bounded from above for some p = (µl, ... , p ) with pt > pi for each i. For u e M and a > 0,

/

4)(ua) = a I 1 - F(u_))

(10.38)

-00 as a - 00

by (10.10) and (10.32). By (10.38), (10.8), and (10.37), (10.39)

da

(4)(u0))

= 1 - a-' (F'(u.), u./p) = a-1 [4)(u.)

- (F'(ua), u./p - u./A) + Hµ(ua)]

10.3. CLASSIFICATION OF SYSTEMS

131

and hence all critical values of 4) are greater than or equal to

ao := inf [ (F'(u), u/p - u/p) - HH(u)] 5 0. Lemma 10.20. For each a < ao, there is a C1-map Aa.: M -> (0, oo) such that ,Da= {ua: u EM, a> Aa(n)}^- M. PROOF. We have 4)(u.) < a for all sufficiently large a > 0 by (10.38), and

(ua)
:

da {4)(u,)) < 0

by (10.39), so there is a unique Aa(u) > 0 such that

4)(ua) > (resp. =, <) a,

a < (resp. =, >) Aa(u)

and the map Aa is Cl by the implicit function theorem. Then W\ {0}, which is

M, deformation retracts to r = {ua : u E M, a > Aa(u)} via (W\{0}) x [0,1]

W\{0},

('a t) H

v1-t+t aui Aa(],)'

71 E (W\{ 0})\ a

u,

uE V

where au = 12(u) and u = uaui.

Lemma 10.21. 4) satisfies (PS). PROOF. If((ui) is al(PS) sequence, then (Ai2) gives

co M

\Pi

lti /

ui I

jIlry

'

<-

i=1

= D(ui)

PIP

1) Pi

C AP;ui,ui

7

- (4)'(ui), ui/p) + Hu(u2)

o(Iluill) + 0(1),

so (ui) is bounded and hence has a convergent subsequence. We can now prove

Theorem 10.22. Assume dim W = oo, (Ail) - (Ai3) and (A4) hold, J0 E with Jo compact, Ao 0 a(Ap, Jo), (10.35), (10.36), and (10.32) hold, and F(u) and (F'(u),u/p - u/µ) are bounded from below and HM(u) is bounded from above for some p with pi > pi for each i. Then (10.1) has a solution

u#0.

10- ANISOTROPIC SYSTEMS

132

PROOF. By Proposition 10.13, Cq(4, 0) # 0 for some q. Since any a < a0 is less than all critical values and 4) is unbounded from below by (10.38), Cq(,p, 00)

Hq-I(.p°)

by Proposition 3.15 (ii)

H9-I(M)

by Lemma 10.20

Hq-I(S)

by Lemma 10.6

=0 by Example 2.5. So the conclusion follows from Proposition 3.28 (ii).

Example 10.23. In problem (10.5), let p = (pI, ... , µ,,,) with pi > pi for each i, set

In

r,,(n) :_

luilu=, e=1

and assume that V. F(x, u) . (u/p - u/p) is bounded from below and (10.40) r, (u) > R 0 < F(x, u) < V. F(x, u) - u/µ for some R > 0. Since Vu F(x, u), and hence also F(x,u), is bounded on bounded sets by (10.6), it then follows that F(u) and

(F'(u), u/p - u/p) = fn VuF(x,u). (u/p-n/p) are bounded from below. The group G = IR\ {0} also has the representation a ul, ... , Ial I I m_1 a um) Pl/v(a)n = (Ial on Rm. If rv(u) > R, set au := rv(u)/R > 1 and u := pl/v(au1) u, and note l/µi-1

that r,,(u) = aulrv(u) = R. Then for a > 1, rv(p11 (a) v) = arv(v.) _ a R > R and hence

F d (F(x, P1/µ(a) u)) _ d (x, PI/v(a) a) aI/v: da rui =l

1

a /Pi

= U-' Vu F(x, P1/v(a) u) . Pl/a(a) it//l > a-' F(x, Pl/v(a) u) by (10.40), and integrating this from a = 1 to au gives F(x, u) > F(x, u) au. So

F(x, u) > c(x) r,,,(u) - C V (x, u) where c (x)

and C > 0, and hence F(u°)

a

i=1

avi/rs-1

=

mn ,r(u)=R i

F(x'u) > 0

R

f nc(x) luilµ` -

C ICI

-00, u e M.

10.3. CLASSIFICATION OF SYSTEMS

Finally, since H, (x, u)

F(x,u) - V F(x, u) -u/µ

is < 0 for r0(u) > R by (10.40) and bounded for rµ(u) < R,

Hµ(x,u)

Hp(u) = fn

is bounded from above.

133

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Titles in This Series 161 Kanishka Perera, Ravi P. Agarwal, and Donal O'Regan, Morse theoretic aspects of p-Laplacian type operators, 2010

160 Alexander S. Kechris, Global aspects of ergodic group actions, 2010

159 Matthew Baker and Robert Rumely, Potential theory and dynamics on the Berkovich projective line, 2010

158 D. R. Yafaev, Mathematical scattering theory: Analytic theory, 2010 157 Xia Chen, Random walk intersections: Large deviations and related topics, 2010 156 Jaime Angulo Pava, Nonlinear dispersive equations: Existence and stability of solitary and periodic travelling wave solutions, 2009

155 Yiannis N. Moschovakis, Descriptive set theory, 2009 154 Andreas Cap and Jan Slovak, Parabolic geometries I: Background and general theory, 2009

153 Habib Ammari, Hyeonbae Kang, and Hyundae Lee, Layer potential techniques in spectral analysis, 2009

152 J6nos Pach and Micha Sharir, Combinatorial geometry and its algorithmic applications: The Alcala lectures, 2009

151 Ernst Binz and Sonja Pods, The geometry of Heisenberg groups: With applications in signal theory, optics, quantization, and field quantization, 2008

150 Bangming Deng, Jie Do, Brian Parshall, and Jianpan Wang, Finite dimensional algebras and quantum groups, 2008

149 Gerald B. Folland, Quantum field theory: A tourist guide for mathematicians, 2008

148 Patrick Dehornoy with Ivan Dynnikov, Dale Rolfsen, and Bert Wiest, Ordering braids, 2008

147 David J. Benson and Stephen D. Smith, Classifying spaces of sporadic groups, 2008 146 Murray Marshall, Positive polynomials and sums of squares, 2008 145 Tuna Altinel, Alexandre V. Borovik, and Gregory Cherlin, Simple groups of finite Morley rank, 2008

144 Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci flow: Techniques and applications, Part II: Analytic aspects, 2008 143 Alexander Molev, Yangians and classical Lie algebras, 2007 142 Joseph A. Wolf, Harmonic analysis on commutative spaces, 2007 141 Vladimir Maz'ya and Gunther Schmidt, Approximate approximations, 2007

140 Elisabetta Barletta, Sorin Dragomir, and Krishan L. Duggal, Foliations in Cauchy-Riemann geometry, 2007

139 Michael Tsfasman, Serge Vladut, and Dmitry Nogin, Algebraic geometric codes: Basic notions, 2007

138 Kehe Zhu, Operator theory in function spaces, 2007 137 Mikhail G. Katz, Systolic geometry and topology, 2007 136 Jean-Michel Coron, Control and nonlinearity, 2007

135 Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci flow: Techniques and applications, Part I: Geometric aspects, 2007

134 Dana P. Williams, Crossed products of C*-algebras, 2007 133 Andrew Knightly and Charles Li, Traces of Hecke operators, 2006 132 J. P. May and J. Sigurdsson, Parametrized homotopy theory, 2006 131 Jin Feng and Thomas G. Kurtz, Large deviations for stochastic processes, 2006 130 Qing Han and Jia-Xing Hong, Isometric embedding of Riemannian manifolds in Euclidean spaces, 2006

129 William M. Singer, Steemod squares in spectral sequences, 2006

128 Athanassios S. Fokas, Alexander R. Its, Andrei A. Kapaev, and Victor Yu. Novokshenov, Painleve transcendents, 2006

TITLES IN THIS SERIES

127 Nikolai Chernov and Roberto Markarian, Chaotic billiards, 2006 126 Sen-Zhong Huang, Gradient inequalities, 2006 125 Joseph A. Cima, Alec L. Matheson, and William T. Ross, The Cauchy Transform, 2006

124 Ido Efrat, Editor, Valuations, orderings, and Milnor K-Theory, 2006

123 Barbara Fantechi, Lothar Gottsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, and Angelo Vistoli, Fundamental algebraic geometry: Grothendieck's FGA explained, 2005

122 Antonio Giambruno and Mikhail Zaicev, Editors, Polynomial identities and asymptotic methods, 2005

121 Anton Zettl, Sturm-Liouville theory, 2005 120 Barry Simon, Trace ideals and their applications, 2005 119 Tian Ma and Shouhong Wang, Geometric theory of incompressible flows with applications to fluid dynamics, 2005

118 Alexandru Buium, Arithmetic differential equations, 2005 117 Volodymyr Nekrashevych, Self-similar groups, 2005 116 Alexander Koldobsky, Fourier analysis in convex geometry, 2005 115 Carlos Julio Moreno, Advanced analytic number theory: L-functions, 2005 114 Gregory F. Lawler, Conformally invariant processes in the plane, 2005

113 William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, and Jeffrey H. Smith, Homotopy limit functors on model categories and homotopical categories, 2004

112 Michael Aschbacher and Stephen D. Smith, The classification of quasithin groups II. Main theorems: The classification of simple QTKE-groups, 2004

111 Michael Aschbacher and Stephen D. Smith, The classification of quasithin groups I. Structure of strongly quasithin K-groups, 2004

110 Bennett Chow and Dan Knopf, The Ricci flow: An introduction, 2004 109 Gore Shimura, Arithmetic and analytic theories of quadratic forms and Clifford groups, 2004

108 Michael Farber, Topology of closed one-forms, 2004 107 Jens Carsten Jantzen, Representations of algebraic groups, 2003 106 Hiroyuki Yoshida, Absolute CM-periods, 2003

105 Charalambos D. Aliprantis and Owen Burkinshaw, Locally solid Riesz spaces with applications to economics, second edition, 2003

104 Graham Everest, All van der Poorten, Igor Shparlinski, and Thomas Ward, Recurrence sequences, 2003 103 Octav Cornea, Gregory Lupton, John Oprea, and Daniel Taxing,

Lusternik-Schnirelmann category, 2003

102 Linda Bass and John Radcliffe, Spatial deterministic epidemics, 2003 101 Eli Gleaner, Ergodic theory via joinings, 2003 100 Peter Duren and Alexander Schuster, Bergman spaces, 2004

99 Philip S. Hirschhorn, Model categories and their localizations, 2003

98 Victor Guillemin, Viktor Ginzburg, and Yael Karshon, Moment maps, cobordisms, and Hamiltonian group actions, 2002

97 V. A. Vassiliev, Applied Picard-Lefschetz theory, 2002

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The purpose of this hook is to present it Morse theoretic study of a very general class of honuigeneous operators that includes the p-Laplacian as a special case. The p-Laplacian operator is a quasilinear differential operator that arises in many applications such as non-Newtonian fluid flows and turbulent filtration in porous media. Infinite dimensional Morse theory has been used extensively to study semilinear problems, but only rarely to study the p-Laplacian. The standard tools of Morse theory for computing critical groups. such as the Morse lemma. the shifting theorem. and various linking and local linking theorems based on cigenspaces, do not apply to quasilinear problems where the Euler functional is nit defined on It I-filbert space or is not C= or adhere there are no eigenspaces to work with.

Moreover. a complete description of the spectrum of a quasilinear operator is generally not available, and the standard sequence of cigenvalties based on the genus is not useful for obtaining nontiis ial critical groups or fur constructing linking sets and local linkings. However, one of the main points of this hook is that the lack of a complete list of ci<_envalues is not an insurmountable obstacle to applying critical point theory. Working with a new sequence of eigcnvalues that uses the cohoniological index, the authors svstcniatically develop alternatise tools such as nonlinear linking and local splitting theories in order to effectively apply Morse theory to quasilinear problems. They obtain nontrivial critical groups in nonlinear cigenvalue problems and use the stability and piercing properties of the coliotnological index to construct new linking sets and local splittings that are readily applicable here.

U

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