NON-TOPOLOGICAL N-VORTEX CONDENSATES FOR THE SELF-DUAL CHERN-SIMONS THEORY MARGHERITA NOLASCO
Abstract. We prove the existence of non-topological N -vortex solutions for an arbitrary number N of vortex points for the self-dual Chern-Simons-Higgs theory with ’t Hooft ”periodic” boundary conditions. We use a shadowing type lemma to glue together any number of single vortex obtained as a perturbation of a radially symmetric entire solution of the Liouville equation.
1. Introduction In recent years the Chern-Simons vortex theory has been extensively studied, in view of its possible application to the physics of high critical temperature superconductivity (see [8] and references therein). In particular, Hong-Kim-Pac [9] and Jackiw-Weinberg [10] have proposed a self-dual model whose electrodynamics, governed only by the Chern-Simons term, produces multi-vortices which are charged both electrically and magnetically. The model is described on the (2+1)-Minkowski space (R1,2 , g), (the metric tensor g = diag[1, −1, −1] is used in the usual way to lower and raise indices) by the following Lagrangean density: 1 1 L(A, φ) = Dη φDη φ + kα,β,γ Fα,β Aγ − 2 |φ|2 (1 − |φ|2 )2 4 k where, A = −iAα dxα , (Aα : R1,2 → R smooth, α = 0, 1, 2) is the gauge potential with covariant derivative DA = d − iA, so that, the corresponding curvature FA = − 2i Fαβ dxα ∧ dxβ , Fαβ = ∂α Aβ − ∂β Aα defines the gauge field, and φ : R1,2 → C is the (smooth) Higgs scalar, with Dη φ = ∂η φ − iAη φ, η = 0, 1, 2. Furthermore the antisymmetric Levi-Civita tensor αβγ is fixed by setting 0,1,2 = 1 and k > 0 is the Chern-Simons coupling parameter. Vortex-condensates are the stationary solutions of the Euler-Lagrange equation corresponding to L, with finite energy, and satisfying suitable gauge invariant periodic boundary conditions introduced by ‘t Hooft [16] (see (1.1)). More precisely, consider the periodic cell domain: a a b b Ω = {x = (x1 , x2 ) ∈ R2 | − ≤ x1 ≤ , − ≤ x2 ≤ } 2 2 2 2 whose boundary is described in terms of the linearly independent vectors e1 = (a, 0) and e2 = (0, b) as follows, ∂Ω = Γ1 ∪ Γ2 ∪ {e1 + Γ2 } ∪ {e2 + Γ1 } ∪ {0, e1 , e2 , e1 + e2 }, with 1 (se1 − e2 ) |s| < 1} 2 1 Γ2 ={x ∈ R2 | x = (se2 − e1 ) |s| < 1}. 2 Γ1 ={x ∈ R2 | x =
Supported by M.U.R.S.T. Cofin: Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari ”. 1
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As proposed by ’t Hooft [16] we require that in Ω the multi-vortex (A, φ) satisfies the following boundary conditions: iξ (x+e ) k e k φ(x + ek ) = eiξk (x) φ(x) A0 (x + ek ) = A0 (x) (1.1) (A j = 1, 2 j + ∂j ξk )(x + ek ) = (Aj + ∂j ξk )(x) 1 2 k x∈Γ ∪Γ \Γ k = 1, 2 where ξ1 and ξ2 are smooth functions defined on a neighborhood of Γ2 ∪ {e1 + Γ2 } and Γ1 ∪ {e2 + Γ1 } respectively. In view of (1.1), to any vortex condensate we can associate an integer N ∈ Z, the vortex number, corresponding to the phase shift of φ around ∂Ω (see also [2] and [18]). More precisely, we have: ξ1 (1, 0+ ) − ξ1 (0, 0+ ) + ξ2 (0+ , 0) − ξ2 (0+ , 1)+ + ξ1 (0, 1− ) − ξ1 (1, 1− ) + ξ2 (1− , 1) − ξ2 (1− , 0) = 2πN. R Consequently, by means of (1.1), for the “magnetic flux” Φ = Ω F12 we obtain a ”quantization effect”, namely Φ = 2πN . In this situation, the vortex number N has the (topological) interpretation of counting the number of zeroes (according to their multiplicity) of the Higgs scalar φ on Ω. Furthermore, the variation of L with respect to the component A0 , gives the Gauss equation governing the system which, for a static solution Aα = Aα (x1 , x2 ), α = 0, 1, 2, and φ = φ(x1 , x2 ), reduces to: kF1,2 = −2A0 |φ|2 ≡ J0 . R Hence for the ”electric charge” Q = Ω J0 we obtain the following ”quantization rule” Q = kΦ = 2πkN . Then, using the boundary conditions (1.1) together with the equation (1.2), it can be shown (see [2]) that the (static) energy relative to the multi-vortex (A, φ) takes the following form: Z Z Z 2 1 k (1.3) E(A, φ) = ( F1,2 + |φ|(|φ|2 − 1))2 + |D1 φ + iD2 φ|2 + F1,2 k Ω 4 |φ| Ω Ω (1.2)
Hence Z (1.4)
E(A, φ) ≥
F1,2 Ω
and the minimum of the energy, at a fixed vortex number N , is saturated by vortexcondensates satisfying the following self-dual problem: D1 φ + iD2 φ = 0 (a) F1,2 + k22 |φ|2 (|φ|2 − 1) = 0 (b) (1.5) kFR1,2 + 2A0 |φ|2 = 0 (c) F = 2πN. (d) Ω 1,2 By direct check, it follows that the viceversa is also true, in the sense that any solution (A, φ) of (1.5) with boundary conditions (1.1) gives rise to a vortex condensate with vortex number equal to N and satisfying: Z Z (1.6) E(A, φ) = 2πN Φ= F1,2 = 2πN Q= J0 = 2πkN. Ω
Ω
For this reason, we shall refer to solution of (1.5) with boundary conditions (1.1) as to a N -vortex condensate.
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Without loss of generality we can limit our attention to the case N > 0, since N = 0 is trivial (it describes the vacua states defined by |φ| = 1 and φ = 0) and N < 0 can be derived from N > 0 by taking complex conjugates. In view of equation (1.5)-(b) and the constraint (1.5)-(d), since the Higgs scalar field must satisfy |φ| ≤ 1, it is reasonable to expect two type of (energy minimizer) N- vortex condensates with the following properties. For given vortex points {p1 , . . . , ps } ⊂ Ω, with integer multiplicity mj such Pevery s that j=1 mj = N , and given k > 0 sufficiently small, the corresponding N-vortex condensate φk satisfies (a) |φk | < 1 and vanishes exactly at pj with multiplicity mj , j = 1, . . . , s; (b - Type I) - (”topological” N-vortex condensates) : |φk | → 1 uniformly on compact sets of Ω \ {p1 , . . . , ps }, as k → 0+ ; or (b - Type II) - (”non-topological” N-vortex condensates): |φk | → 0 uniformly on compact sets of Ω \ {p1 , . . . , ps }, as k → 0+ ; Ps (c) F1,2 (φk , Ak ) → 2π j=1 mj δpj in the sense of measure, as k → 0+ (where δp denotes the Dirac measure with pole at p). In view of the property (c), we shall refer to the zeroes of the Higgs scalar as to the vortex points. It is shown in [17] that vortex condensates of ”type I” exist for all prescribed vortex number N and correspond to the “maximal” N -vortex condensates found by Caffarelli and Yang in [2]. Note that vortex solutions of ”type I” have the same behavior of the Ginzburg- Landau vortices (see [18]). On the other hand, the vortex condensates of ”type II” have not been well understood yet. In [17] the author proves the existence of a solution (1.5) with b.c. (1.1) with a single vortex-point, such that |φk | → 0, as k → 0+ , in C q (Ω) (for any q > 0) and F1,2 (φk , Ak ) is a compact sequence in L1 (Ω). Later, in [11] the authors establish the existence of double-vortex condensates such that |φk | → 0, as k → 0+ , in C 0 (Ω) and either F1,2 (φk , Ak ) is a compact sequence in L1 , as k → 0+ ; or, for any sequence kn → 0, there exists a point p ∈ Ω with p 6= p1 , p2 (but completely characterized in terms of the vortex points) such that: F1,2 (φkn , Akn ) → 4πδp in the sense of measures. We mention here that this last case in fact occurs when the torus T 2 (namely periodic b.c.) is replaced by the two-sphere S 2 . This result for S 2 -double vortex condensates was obtained before by Ding-Jost-Li-Wang [7]. More interestingly, in [7] the authors obtain, in addition, the existence of a second double-vortex which satisfies |φk | → 0, as k → 0+ , in C q (Ω) (for any q > 0) and F1,2 (φk , Ak ) is a compact sequence in L1 (Ω) for any assigned pair of distinct vortex-points. Let us mention also that in [5] (see also [6] for the case N = 2), the authors have established the existence of a multi-vortex (Ak , φk ) with an arbitrary number of vortex, such that φk → 0 point-wise a.e. in Ω as k → 0+ . Therefore, up to these results the question of the existence of N-vortex condensates satisfying (a)-(b - Type II) and (c) is still open. The aim of this paper is to give a complete positive answer to this question. More precisely, we prove the following result: Theorem. For any vortex number N ∈ N there exists k¯ > 0 such that for any ¯ and every given vortex points {p1 , . . . , ps } ⊂ Ω, with integer multiplicity k ∈ (0, k)
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M. NOLASCO
Ps mj such that j=1 mj = N , there exists a N -vortex condensates (φk , Ak ) satisfying (1.1), (1.5), (1.6) and the following properties: (a) |φk | < 1 and vanishes exactly at pj with multiplicity mj , j = 1, . . . , s; q + (b) |φk | → 0 Cloc (Ω \ {p P1s, . . . , ps }), ∀q ≥ 0, as k → 0 ; (c) F1,2 (φk , Ak ) → 2π j=1 mj δpj in the sense of measure, as k → 0+ . 3 Our proof relies in a perturbation method together with a gluing technique which allows us to give a rather precise behavior of the N-vortex solutions |φk | (for k > 0 sufficiently small) near each vortex point pj (with multiplicity mj ), in terms of ρj (x) =
8(mj +1)2 |x|2mj (1+|x|2mj +2 )2
which is a solution of the Liouville equation −∆ ln ρ = ρ − 4πmj δx=0
in R2 .
In fact, for any > 0 sufficiently small there exists δ → 0+ , as → 0+ , such that for k = δ and for x sufficiently close to pj 1 |φk (x)|2 = 2 ρj ( |x − pj |) + o(1), as → 0+ . δ Let us point out that the problem of finding N -vortex condensates of type I and II bares strong analogies with the search of the topological and non-topological solutions for the self-dual Chern-Simons-Higgs theory on R2 , where one replaces the periodic boundary conditions (1.1) with suitable decay conditions at infinity. Over R2 , topological and non-topological vortices have been established in [19], [15] and [14] (radial case), and [3]. In fact, we follow closely the construction of [3] to derive the local behavior of our solutions near each vortex point. Then, we use a version of shadowing lemma for elliptic PDE (see [1]) to glue together such a local solutions to obtain the desired N-vortex condensates. In concluding, let us mention that the problem of existence of topological and non-topological vortex solutions has been studied also for non-abelian Chern-Simons Higgs theory ([21], [20] and [12]) and for the Maxwell Chern-Simons-Higgs system ( [4] and [13]). Acknowledgements: The author wishes to thank G. Tarantello for useful discussions. 2. Preliminaries We shall follow an approach introduced by Taubes [18] for the study of self-dual Ginzburg-Landau vortices (we refer to [2] for the case of Chern-Simons Higgs self dual equation). Let us rewrite the first equation in (1.5) as follows, (2.1) −2i∂¯ ln φ = A1 + iA2 . From (2.1) we may conclude that φ must have only point-zeroes with integer multiplicity. Thus, if φ vanishes at pj with multiplicity mj ∈ N (j = 1, . . . , s) then differentiating the equation (2.1) we get 1 F1,2 = − ∆ln |φ|2 on Ω \ {p1 , . . . ps } 2 and, by means of the equation (b) in (1.5), we obtain that u = ln |φ|2 must satisfies the following nonlinear elliptic equation: s X −∆u = 42 eu (1 − eu ) − 4π mj δp in Ω j k (2.2) j=1 u doubly periodic on ∂Ω.
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Viceversa, it can be shown that , if u satisfy (2.2), then by defining: s X 1 mj arg (x − pj )}, φ(x) = exp{ u(x) + i 2 j=1
we may derive A1 , A2 from (2.1) and A0 from (1.5)-(c) to obtain a N -vortex condensate satisfying (1.1)-(1.5) with vortex-points pj ∈ Ω\∂Ω (by periodicity, w.l.o.g. we choose Ps pj 6∈ ∂Ω, ∀j),and relative multiplicity mj ∈ N (j = 1, . . . , s) such that N = j=1 mj (see [18]). As a first step we extend by periodicity equation (2.2) to the entire R2 , namely s X X −∆u = 42 eu (1 − eu ) − 4π mj δp(n1 ,n2 ) in R2 k j (2.3) 2 (n1 ,n2 )∈Z j=1 u(x + ek ) = u(x) k = 1, 2 ∀x ∈ R2 (n ,n )
where the points pj 1 2 = pj + n1 ek + n2 e2 , j = 1, . . . s, (n1 , n2 ) ∈ Z2 , define a periodic lattice of vortex points with double periodicity e1 = (a, 0), e2 = (0, b). We set k = 2δ > 0. As it would be clear later, we will consider > 0 as a perturbation parameter and δ > 0 as a scaling parameter. Then, for u solution of (2.3) we set u ˆ(x) = u(δx) − ln 2 , where u ˆ satisfies the equation u ˆ
2 u ˆ
−∆ˆ u = e (1 − e ) − 4π
(2.4)
s XX
mj δpˆnj
in R2
n∈Z2 j=1 pn j
where pˆnj = δ for j = 1, . . . , s, mj ∈ N and n ≡ (n1 , n2 ) ∈ Z2 . To simplify the notation, we set pˆn0 ≡ (n1 eˆ1 + n2 eˆ2 ), with eˆi = 1δ ei (for i = 1, 2) and m0 ≡ 0. The idea is to look first at the local behavior of the solution of the equation (2.4) around each vortex point searching the radially symmetric solutions of the equation for mj -vortex concentrated at each point pˆnj . 3. Radially symmetric case: single vortex point In this section we briefly sketch a result contained in [3] (see also [14]). We consider the problem of the existence of radially symmetric solutions of the following equation for N-vortex concentrated at the origin of R2 (eventually N = 0), : (3.1)
( −∆u = eu (1 − 2 eu ) − 4πN δ0 u(x) → −∞ as |x| → +∞.
in R2 ,
Note that by the maximum principle a solution of (3.1) satisfies (3.2)
2 eu(x) < 1
∀x ∈ R2 .
The idea is to construct a solution close to the radial function ln ρN where ρN (x) is given by 8(N + 1)2 |x|2N (1 + |x|2N +2 )2 which is a solution of the Liouville equation (3.3)
(3.4)
ρN (x) =
−∆ ln ρ = ρ − 4πN δ0
in R2 .
Following [3], we introduce also the function wN ∈ C 2 (R+ ) which is the radial solution of the following equation (3.5)
−∆w = ρN (|x|)w − ρ2N (|x|)
in R2 ,
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M. NOLASCO
and we make a change of variable in the equation (3.1) as follows: u(|x|) = ln ρN (|x|) + 2 wN (|x|) + 2 v(|x|).
(3.6)
If u is a radial solution of equation (3.1) then the radially symmetric function v solves the following equation: 2 2 1 1 ρ e (v+wN ) − ρ2N e2 (v+wN ) − 2 ρN − ρN wN + ρ2N . 2 N To find a radial solution of (3.7) we introduce the following functional setting. We define the inner products Z (u, v)Y = (1 + |x|2+α )uv dx 2 R Z (3.8) uv dx, (u, v)X =(∆u, ∆v)Y + 2+α ) R2 (1 + |x|
(3.7)
−∆v =
with α ∈ (0, 12 ) and we consider the Hilbert spaces (3.9)
2,2 X ={u ∈ Wloc (R2 ) : kuk2X = (u, u)X < +∞}
Y ={u ∈ L2loc (R2 ) : kuk2Y = (u, u)Y < +∞}.
We denote by X r and Y r the Hilbert spaces of radially symmetric functions of X and Y , respectively. In [3]-lemma 1.1 and lemma 2.2, the authors prove that (3.10)
|v(x)| ≤ kvkX (ln+ |x| + 1)
∀x ∈ R2
∀v ∈ X;
and (3.11)
|wN (|x|)| ≤ C(ln+ |x| + 1) ∀x ∈ R2 C as |x| → +∞. |∇wN (|x|)| ≤ |x|
Now, let us introduce the map PN : B ⊂ X r × R → Y r , where the domain is a suitable bounded subset of X r × R , and 2 2 1 1 ρN e (v+wN ) − ρ2N e2 (v+wN ) − 2 ρN − ρN wN + ρ2N . 2 In view of (3.10) and (3.11), it is easy to check that the map PN is well defined and smooth on any bounded set of X r , provided || ≤ 0 with 0 sufficiently small. Now, if a couple (v, ) solves the functional equation PN (v, ) = 0 then v is in fact a solution of the equation (3.7) for the corresponding value of . Since PN (0, 0) = lim→0 PN (, 0) = 0, the idea is to apply the implicit function theorem to the map PN near the origin. Thus we consider the bounded linear operator
(3.12)
(3.13)
PN (v, ) = ∆v +
LrN = Dv PN (0, 0) = ∆ + ρN : X r → Y r .
We have Lemma 3.1. (see [3]) The operator LrN : X r → Y r is onto and Ker LrN = span {φN }, where φN (|x|) =
1−|x|2N +2 . 1+|x|2N +2
In view of lemma 3.1 we introduce the Hilbert space HNr ⊂ X r given by Z uφN r r (3.14) HN = {u ∈ X : dx = 0} 2+α R2 1 + |x|
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Then, by lemma 3.1 and the implicit function theorem, for > 0 sufficiently ∗ ∗ ∗ small, there exists v, ∈ HNr such that PN (v, , ) = 0 and kv, k → 0, as → 0. N N N X ∗ By standard regularity theory, v,N is smooth and, by (3.10) we have ∗ |v, (|x|)| ≤ C()(ln+ |x| + 1) N
(3.15)
∗ with C() = kv, k → 0 as → 0. N X ∗ Moreover, since |x||∆v, | → 0 as |x| → +∞, it is easy to check that N ∗ |∇v, (|x|)| ≤ N
(3.16)
C |x|
as |x| → +∞.
In view of (3.11) and (3.15) the radial solution of (3.1) given by ∗ u∗,N (|x|) = ln ρN (|x|) + 2 wN (|x|) + 2 v, (|x|) N
(3.17)
is in fact close to the solution of the Liouville equation (in the topology of X and 0 in Cloc (R2 )). Moreover, we have the following decay estimate (see [3] - (3.6)) : ∗
eu,N (|x|) = O(
(3.18)
1 ) |x|2N +4+β
as |x| → +∞
with 0 < β = β() → 0 as → 0, and by the maximum principle we have also ∗
2 eu,N (x) < 1
(3.19)
∀x ∈ R2
4. Linear properties of the radial single-vortex solution In this section we study the properties of the linear bounded operator (4.1)
∗ A,N ≡ Dv PN (v, , ) : X → Y N
(for > 0 sufficiently small) given by (4.2)
A,N = ∆ + ρN e
2
∗ ) (wN +v, N
− 22 (ρN )2 e2
2
∗ ) (wN +v, N
By Taylor’s expansion and in view of (3.15) and (3.11), we have as → 0 (in the norm operator) (4.3)
A,N = LN + 2 (ρN wN − 2ρ2N ) + o(2 )
where LN = ∆ + ρN : X → Y . The operator LN : X → Y is studied in [3]. In particular, in [3]- lemma 2.4 and proposition 2.2 the Kernel and the Image of LN are completely characterized as follows. Let consider the following functions (for x = (|x| cos θ, |x| sin θ)) 1 − |x|2N +2 1 + |x|2N +2 |x|N +1 cos(N + 1)θ φ+,N (x) = 1 + |x|2N +2 N +1 |x| sin(N + 1)θ φ−,N (x) = . 1 + |x|2N +2 φN (x) =
(4.4)
We have, Lemma 4.1. (see [3]) Let consider the operator LN : X → Y . Then, (4.5)
(i) Ker LN = span{φN , φ+,N , φ−,N } Z (ii) Im LN = {f ∈ Y : f φ±,N dx = 0} R2
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M. NOLASCO
Moreover, in [3]- lemma 2.5 is proved the following inequality: Z (4.6) (wN ρN − 2ρ2N )φ2±,N dx < 0. R2
R 2π R 2π R 2π On the other hand, since 0 sin(N + 1)θ dθ = 0 cos(N + 1)θ dθ = 0 sin(N + 1)θ cos(N + 1)θ dθ = 0 we get Z (4.7)
R2
Z R2
(wN ρN − 2ρ2N )φ±,N φ∓,N dx = 0 (wN ρN − 2ρ2N )φN φ±,N dx = 0.
Now, we introduce the Hilbert space HN ⊂ X given by Z uφN (4.8) HN = {u ∈ X : dx = 0}. 2+α R2 1 + |x| We have Lemma 4.2. There exists 0 > 0 and a constant C > 0 (independent on ) such that for any ∈ (0, 0 ) and for any v ∈ HN we have (4.9)
kA,N vkY ≥ C2 kvkX .
Proof. Arguing by contradiction we suppose that there exists a sequence n → 0, as n → +∞, and a sequence vn ∈ HN with kvn kX = 1 such that 12 kAn ,N vn kY → 0, n as n → +∞. Hence, in particular kAn ,N vn kY = o(2n ) and vn → v¯, weakly in X, as n → +∞, and, by Sobolev embedding theorem, strongly in Lploc (R2 ), for any p ≥ 2. Moreover, it is easy to check that An ,N vn → LN v¯ weakly in Y , as n → +∞ . Hence LN v¯ = 0 and by lemma 4.1, (4.10)
v¯ = c0 φN + c+ φ+,N + c− φ−,N
for some constant c0 , c+ , c− ∈ R. Now, we want to prove that at least one of these constants is not equal to zero, namely that v¯ 6≡ 0. In fact, we prove that vn → v¯ strongly in X. Note that, by (4.3) we have (4.11)
k∆vn − ∆¯ v kY ≤ kρN (vn − v¯)kY + o(1)
as n → +∞.
Since vn → v¯ strongly and since |vn (x)| ≤ kvn kX (ln+ |x| + 1) (see (3.10)), by decomposing R into a compact set and its complementary, and using the Rellich-Kondrachev compactness theorem for the compact set, we may conclude v that kρN (vn − v¯)kY → 0. With a similar argument we get also k vn −¯ 1 kL2 → 0, 2+α in L2loc (R2 ) 2
(1+|x|
)2
and hence we may conclude kvn − v¯kX → 0, namely v¯ 6≡ 0. Moreover, since R vn φN dx = 0, passing to the limit as n → +∞ we get v¯ ∈ HN . Then, in R2 1+|x|2+α R φ±,N φN particular, since R2 1+|x| 2+α dx = 0 we get c0 = 0 and (4.12)
v¯ = c+ φ+,N + c− φ−,N .
with c2+ + c2− 6= 0. v ¯ Now we reach a contradiction as follows. We set w ¯ = (1+|x| 2+α ) ∈ Y . Then, we have that 1 o(2n ) (4.13) (A v , w) ¯ ≤ kwk ¯ Y →0 as n → +∞. , n Y n N 2n 2n On the other hand, in view of (4.3) we have 1 1 (4.14) 2 (An ,N vn , w) ¯ Y = 2 (LN vn , v¯)L2 (R2 ) + ((wN ρN − ρ2N )vn , v¯)L2 (R2 ) + o(1) n n
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Now, we claim that (LN u, φ±,N )L2 (R2 ) = 0, for any u ∈ X. Indeed, let m ∈ C ∞ (R+ ) be such that m(r) = 1 for r ∈ [0, 1] and m(r) = 0 for r > 2. We denote mλ (x) = m(λx). By H¨ older inequality, we have
mλ φ±,N − φ±,N
(4.15) (LN u, (mλ φ±,N − φ±,N ))L2 (R2 ) ≤ kLukY 1 (1 + |x|2+α ) 2 L2 (R2 ) and, by dominated convergence, it is easy to check that
mλ φ±,N − φ±,N
(4.16) →0 as λ → 0.
(1 + |x|2+α ) 12 L2 (R2 ) Therefore, by integration by parts and in view of lemma 4.1, we may conclude that, for any u ∈ X, (4.17) (LN u, φ±,N )L2 (R2 ) = lim (LN u, mλ φ±,N )L2 (R2 ) = λ→0 Z = lim [λ2 (∆m)(λx)φ±,N u + 2(λ(∇m)(λx)∇φ±,N )u] dx = 0. λ→0
R2
Finally, passing to the limit as n → +∞ in (4.14), in view of the inequality (4.6), we get the following contradiction Z 1 2 0 = lim 2 (An ,N vn , w) ¯ Y = c+ (wN ρN − 2ρ2N )φ2+,N + n→+∞ n R2 Z (4.18) 2 + c− (wN ρN − 2ρ2N )φ2−,N < 0. R2
In view of lemma 4.2 we may conclude that for > 0 sufficiently small, the operator A,N : HN → Y is injective, namely that Ker A,N = ∅ in HN . Lemma 4.3. For > 0 sufficiently small, let consider the operator A,N : HN → Y . Then, Im A,N is closed in Y . Proof. Fixed > 0 sufficiently small, we consider a convergent sequence fn → f in Y , as n → +∞, with fn = A,N wn and wn ∈ HN , for any n ∈ N. By lemma 4.2 we have (4.19)
kA,N wn kY ≥ C2 kwn kX ,
∀n ∈ N.
Hence wn is a bounded sequence in X. By decomposing R2 into a compact set and its complementary, and using Rellich - Kondrachev theorem for the compact set, it is easy to check that the operator K = A,N − ∆ : HN → Y is a compact operator. Therefore, denoting by gn = fn − Kwn , we have ∆wn = gRn with gn → g (up to 1 ln |x − y|g(y) dy we subsequences) in Y , as n → +∞. Now, denoting w ¯ = 2π R2 have Z 1 | ln |x − y|(gn (y) − g(y)) dy| ≤ 2π R2 Z 12 1 ln2 |x − y| ≤ dy kgn − gkY 2+α 2π R2 1 + |y|
|wn (x) − w(x)| ¯ ≤ (4.20)
We can estimate the last integral as follows
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M. NOLASCO
Z R2
ln2 |x − y| dy = 1 + |y|2+α
Z R2
ln2 |τ | dτ = 1 + |x − τ |2+α ln2 |τ | dτ + 1 + |x − τ |2+α
Z = |τ |≤1
Z |τ |≥1
ln2 |τ | dτ = I1 + I2 1 + |x − τ |2+α
We have Z I1 = |τ |≤1
ln2 |τ | dτ ≤ 1 + |x − τ |2+α
Z
ln2 |τ | dτ ≤ C,
|τ |≤1
and Z I2 = |τ |≥1
ln2 |τ | dτ = 1 + |x − τ |2+α
Z {|τ |≥1}∩{|τ |≥2|x|}
ln2 |τ | dτ + 1 + |x − τ |2+α
2
Z + {|τ |≥1}∩{|τ |<2|x|}
ln |τ | dτ ≤ C(1 + (ln+ 2|x|)2 ). 1 + |x − τ |2+α
Hence, we obtain (4.21)
|wn (x) − w(x)| ¯ ≤ Ckgn − gkY (ln+ |x| + 1),
and, as n → +∞, Z (4.22) kwn − wk ¯ X ≤C R2
12 (ln+ |x| + 1)2 dx kgn − gkY ≤ Ckgn − gkY → 0. (1 + |x|2+α )
Therefore we may conclude that w ¯ ∈ X and, since A,N is a closed operator, ¯ = f . Moreover, since wn ∈ HN and HN is closed in X, we have that in fact A,N w w ¯ ∈ HN . In view of lemma 4.2 in order to conclude that A,N : HN → Y is invertible we prove: Lemma 4.4. For > 0 sufficiently small, the operator A,N : HN → Y is onto. Proof. By lemma 4.3 we have that Im A,N is closed in Y , hence we can decompose Y as follows: (4.23)
Y = ImA,N ⊕ (Im A,N )⊥ .
For any > 0 sufficiently small, let ξ ∈ (Im A,N )⊥ , with kξ kY = 1. Then, we have (4.24)
∀u ∈ HN .
(A,N u, ξ )Y = 0
Hence, setting ψ (x) = (1 + |x|2+α )ξ (x), we have (4.25)
∀u ∈ HN
(A,N u, ξ )Y = (A,N u, ψ )L2 (R2 ) = 0
By standard density argument and integration by parts, this implies (4.26)
in R2 ,
A,N ψ = 0
for any > 0 sufficiently small. Then, arguing as in the proof of lemma 4.2, we have that ψn → ψ¯ strongly in X, for some sequence n → 0, as n → +∞, and LN ψ¯ = 0. Hence, by lemma 4.1, (4.27) ψ¯ = co φ + c+ φ+, + c− φ−, . N
N
N
On the other hand, as we have already proved in lemma 4.2 (see (4.17)) we have that (LN ψn , φ±,N )L2 (R2 ) = 0.
NON-TOPOLOGICAL N-VORTEX CONDENSATES ...
11
Hence, we get 0= (4.28)
1 (An ,N ψn , φ±,N )L2 (R2 ) =((wN ρN − ρ2N )ψn , φ±,N )L2 (R2 ) + o(1) → 2 Z →c± (wN ρN − 2ρ2N )φ2±,N R2
as n → +∞. Hence by the inequality (4.6) we may conclude that c± = 0 and ψ¯ = c0 φN , with c0 6= 0. ¯ φN ψ ¯ Now, setting ξ¯ = (1+|x| 2+α ) = c0 (1+|x|2+α ) (note that ξ ∈ Y ), by integration by ¯ Y = c0 (L u, φ )L2 (R2 ) = c0 (u, L φ )L2 (R2 ) = 0 for parts we have that (LN u, ξ) N N N N ¯ any u ∈ X, hence ξ ∈ (Im LN )⊥ . On the other hand, since Z ¯ ±, dx = 0 ξφ N R2
by lemma 4.1, we have also that ξ¯ ∈ Im LN , hence we may conclude that necessarily ¯ Y 6= 0. ξ¯ ≡ 0, that is a contradiction since kξk As a direct consequence of lemma 4.2 and lemma 4.4 we get the following result: Proposition 4.5. There exists 0 > 0 such that for any ∈ (0, 0 ) the operator A,N : HN → Y is invertible. Moreover, there exists a constant C > 0 (independent on ) such that for any ∈ (0, 0 ) and for any u ∈ Y we have C (4.29) k(A,N )−1 ukX ≤ 2 kukY . 5. Periodic lattice of radial vortex solutions In this section we simply extend by periodicity to each vortex points pˆnj (j = 0, . . . , s and n = (n1 , n2 ) ∈ Z2 ), with multiplicity mj , all the results obtained for the N-vortex centered at the origin. We introduce the functions ρˆnj : R2 → R and w ˆjn : R2 → R, for j = 0, . . . , s and n ∈ Z2 defined by (5.1)
ρˆnj (x) =ρmj (|x − pˆnj |) =
8(mj + 1)2 |x − pˆnj |2mj (1 + |x − pˆnj |2mj +2 )2
w ˆjn (x) =wmj (|x − pˆnj |), where, for any N ∈ N ∪ {0}, ρN and wN have been introduced in section 3 (see (3.3) - (3.5)), and pˆnj hare the scaled vortex points with integer multiplicity mj (see section 2). For j = 0, . . . , s, let us introduce the following inner products: Z (u, v)Yˆ n = uv(1 + (δ|x − pˆnj |)2+α )δ 2 dx j 2 R Z (5.2) uv (u, v)Xˆ n =(∆u, ∆v)Yˆ n + δ 2 dx j j ˆnj |)2+α ) R2 (1 + (δ|x − p with α ∈ (0, 12 ) and δ > 0 is the scaling parameter already introduced at the end of section 2. Remark 5.1. Note that if supp u ⊂ B δ1 (ˆ pnj ) = {x : |x − pˆnj | ≤ 1δ } then · kukYˆ n ≈ ku( )kL2 (B1 (pnj )) j δ (5.3) · · kuk2Xˆ n ≈ δkuk2H 2 (R2 ) ≈ k(δ 2 ∆u)( )k2L2 (B1 (pn )) + ku( )k2L2 (B1 (pn )) . j j j δ δ
12
M. NOLASCO
ˆ n and Yˆ n as follows: Now, we introduce the Hilbert spaces X j j
(5.4)
ˆ jn ={u ∈ W 2,2 (R2 ) : kuk2ˆ n = (u, u) ˆ n < +∞ } X loc X X j
j
Yˆjn ={u ∈ L2loc (R2 ) : kuk2Yˆ n = (u, u)Yˆ n < +∞}. j
j
∗ For any > 0 sufficiently small, we define = v,m (|x − pˆnj |), for j = j ∗ 0, . . . , s, where v,N is the radial solution of the functional equation PN (v, ) = 0 (see section 3). Hence, in particular if we set n vˆ,j (x)
2 n 1 1 n 2 (wˆjn +v) − (ˆ ρnj )2 e2 (wˆj +v) − 2 ρˆnj − ρˆnj w ˆjn + (ˆ ρnj )2 ρˆ e 2 j n we have Pjn (ˆ v,j , ) = 0, for any > 0 sufficiently small. Moreover, we set φˆnj (x) = φmj (|x − pˆnj |), with φN (|x|) as defined in lemma 3.1, ˆn ⊂ X ˆ n given by and the Hilbert space H j j Z uφˆnj ˆ jn = {u ∈ X ˆ jn : (5.6) H dx = 0}. ˆnj |)2+α ) R2 (1 + (δ|x − p
(5.5)
Pjn (v, ) = ∆v +
n ˆ n → Yˆ n given by Then, considering the operator Aˆn,j = Dv Pjn (ˆ v,j , ) : H j j
(5.7)
2 n n 2 n n Aˆn,j = ∆ + ρˆnj e (wˆj +ˆv,j ) − 22 (ˆ ρnj )2 e2 (wˆj +ˆv,j )
it is easy to check that we may extend the result stated in proposition 4.5 as follows: Proposition 5.2. There exists 0 > 0 such that for any ∈ (0, 0 ), the operator ˆ n → Yˆ n is invertible. Moreover, there exists a constant C > 0 (independent Aˆn,j : H j j on both and δ) such that for any ∈ (0, 0 ) and for any u ∈ Yˆjn we have (5.8)
k(Aˆn,j )−1 ukXˆ n ≤ j
C kukYˆ n . j 2
6. A shadowing-type lemma: functional setting In this section we use a gluing technique introduced in [1] to construct a solution of the equation (2.4) which ”shadows” near each vortex point the local solutions u ˆn,j = u∗,mj (|x − pˆnj |), where u∗,mj is the radially symmetric N-vortex solution obtained in section 3. We set r = 12 mini6=j {|pi − pj |; d(pj , ∂Ω)}, (note that pj 6∈ ∂Ω) and we introduce the sets Bj ={x ∈ Ω : |x − pj | ≤ r} j = 1, . . . , s [ r (6.1) B0 ={x ∈ R2 : d(x, Ω \ Bj ) ≤ }. 2 j=1,...,s Then for any n = (n1 , n2 ) ∈ Z2 , we set (6.2)
Bjn = {x ∈ R2 : x − n1 e1 − n2 e2 ∈ Bj }
j = 0, . . . , s.
The collection of sets {Bjn }, for j = 0, . . . , s and n = (n1 , n2 ) ∈ Z2 is a locally finite (periodic) covering of R2 . We consider an associate partition of unity defined as follows: let 0 ≤ ϕj ∈ Cc∞ (Bj ) (j = 0, . . . , s) and, for n = (n1 , n2 ) ∈ Z2 , let ϕnj (x) = ϕj (x − n1 e1 − n2 e2 ), be such that (6.3)
s XX n∈Z2
j=0
ϕnj (x) = 1,
∀x ∈ R2 .
NON-TOPOLOGICAL N-VORTEX CONDENSATES ...
13
Now, we introduce the scaling parameter δ > 0 as follows. Let define (6.4)
ˆ n = {x ∈ R2 : δx ∈ B n }, B j j
n = (n1 , n2 ) ∈ Z2 ,
j = 0, . . . , s,
ˆ n ) and {ϕˆn }n,j is a partition and ϕˆnj (x) = ϕnj (δx). Hence, in particular ϕˆnj ∈ Cc∞ (B j j of unity relative to the locally finite (periodic) covering of R2 given by the collection ˆ n} of sets {B (j = 0, . . . , s and n = (n1 , n2 ) ∈ Z2 ), with the property that P Ps j n,jn ˆj (x) = 1, for any x ∈ R2 . Note that supx∈Bˆ n {|∇ϕˆnj |} ≤ Cδ and n∈Z2 j=0 ϕ j
supx∈Bˆ n {|∆ϕˆnj |} ≤ Cδ 2 . j We introduce also the sets (6.5)
ˆ n : ϕˆn (x) = 1} Uˆjn = {x ∈ B j j
n = (n1 , n2 ) ∈ Z2 .
j = 0, . . . , s,
Now, we define the following Banach spaces : ˆ δ ={u ∈ W 2,2 (R2 ) : ϕˆnj u ∈ H ˆ jn and kuk ˆ = sup kϕˆnj uk ˆ n < +∞} H loc Hδ X (6.6)
j
n,j
Yˆδ ={u ∈ L2loc (R2 ) : ϕˆnj u ∈ Yˆjn and kukYˆδ = sup kϕˆnj ukYˆ n < +∞}. j
n,j
The idea is to solve the equation (2.4) as a functional equation defined on a ˆ δ with values in Yˆδ , and using a Banach fixed point theorem bounded domain of H in a neighborhood of each radial vortex solution centered at the point pˆnj , provided the vortex points are sufficiently far away one from each others, namely for δ > 0 sufficiently small. Thus, we introduce the following change of variable in the equation (2.4):
u ˆ(x) =
s XX n∈Z2
(6.7) =
ϕˆnj (x)unj (x) + 2 z(x) =
j=0
s XX
n ϕˆnj (x)(ln ρˆnj (x) + 2 w ˆjn (x) + 2 vˆ,j (x)) + 2 z(x).
n∈Z2 j=0
Then u ˆ is a solution of (2.4) if and only if z satisfies the functional equation F,δ (z) = 0, where
F,δ (z) =∆z + (6.8) −
X
n 1 X n n 2 (wˆjn +ˆv,j ) 2 z n ϕˆj ρˆj e (e Rj (x) − 1)− 2 n;j
ϕˆnj (ˆ ρnj )2 e2
2
n (w ˆjn +ˆ v,j )
2
(e2 z (Rjn (x))2 − 1) +
n;j
1 X n C (x) 2 n;j j
with n Cjn (x) = [∆, ϕˆnj ](ln ρˆnj + 2 w ˆjn + 2 vˆ,j )
j = 0, . . . , s,
n = (n1 , n2 ) ∈ Z2
where [∆, ϕˆnj ] = ∆ϕˆnj − ∇ϕˆnj ∇, and for n = (n1 , n2 ) ∈ Z2
(6.9)
s X ρˆn n n R0n (x) = exp{ ϕˆnl [ln nl + 2 (w ˆln − w ˆ0n ) + 2 (ˆ v,l − vˆ,0 )]}× ρˆ0 l=1
× exp{
X
k=<
>
ϕˆk0 [ln
ρˆk0 k n + 2 (w ˆ0k − w ˆ0n ) + 2 (ˆ v,0 − vˆ,0 )]} ρˆn0
14
M. NOLASCO
where we denote by << n >> the nearest neighborhoods of the given n, and, for j = 1, . . . , s Rjn (x) = exp{ϕˆn0 [ln
(6.10)
ρˆn0 n n + 2 (w ˆ0n − w ˆjn ) + 2 (ˆ v,0 − vˆ,j )]}. ρˆnj
Remark 6.1. Note that s XX 2 n n exp[ ϕˆnj (x)ˆ unj (x)] = ρˆnj e (wˆj +ˆv,j ) Rjn (x) n∈Z2 j=0
hence in view of (3.19) we may conclude that 2 ρˆnj e
(6.11)
2
n (w ˆjn +ˆ v,j )
Rjn (x) < 1
∀x ∈ R2 .
Now, we give some estimates in terms of the scaling parameter δ → 0 of the error terms (Rjn (x) − 1) and Cjn (x). We have Lemma 6.2. We have (i) sup ϕˆnj (x)ˆ ρnj (x)|Rjn (x) − 1| = O(δ 4−β() ),
(6.12)
as δ → 0,
x∈R2
with β() → 0, as → 0, and (ii) k
(6.13)
X
Cjn (x)kYˆδ = O(δ 2 | ln δ|)
as δ → 0.
n;j
Proof. (i) Note that for any n = (n1 , n2 ) ∈ Z2 , j = 0, . . . , s, we have ˆ n \ Uˆn ∀x ∈ B j j
C1 δ 2mj +4 ≤ ρˆnj (x) ≤ C2 δ 2mj +4
(6.14)
for some positive constants C1 , C2 . Hence,
sup ˆ n ∩B ˆn x∈B 0 j
(6.15)
sup ˆ k ∩B ˆn x∈B 0 0
ρˆnj (x) ≤ Cδ 2mj ρˆn0 (x) ρˆk0 (x) ≤C ρˆn0 (x)
j = 1, . . . , s;
∀k =<< n >> .
Moreover, by (3.10), (3.11) and (3.15), we have sup ˆ n ∩B ˆn x∈B 0 j
sup ˆ k ∩B ˆn x∈B 0 0
n n (|w ˆ0n (x) − w ˆjn (x)| + |ˆ v,0 (x) − vˆ,j (x)|) ≤C| ln δ|
j = 0, . . . , s;
n k (|w ˆ0n (x) − w ˆ0k (x)| + |ˆ v,0 (x) − vˆ,0 (x)|) ≤C| ln δ|
∀k =<< n >> .
as δ → 0. Therefore in view of the definitions (6.9) and (6.10) and since ϕˆnj (x) = 0 for ˆ n and Rn (x) = 1 for x ∈ Uˆn , we may conclude that as δ → 0 x ∈ R2 \ B j j j (6.16)
sup ϕˆnj (x)ˆ ρnj (x)|Rjn (x) − 1| ≤ C
x∈R2
sup x∈Bjn \Ujn
ρˆnj (x)(Rjn (x) + 1) ≤ Cδ 4−β()
for any j = 0, . . . , s n = (n1 , n2 ) ∈ Z2 , with β() → 0, as → 0.
NON-TOPOLOGICAL N-VORTEX CONDENSATES ...
15
ˆ n \ U n we have (ii) Since supp [∆ϕˆnj (x)], supp[∇ϕˆnj (x)] ⊂ B j j k
X
Cjn (x)kYˆδ ≤ C sup kϕˆnj Cjn (x)kYˆ n ≤ j
n;j
n;j
≤
(6.17)
sup ˆ n \U n x∈B j j
+
sup ˆ n \U n x∈B j j
n |∆ϕˆnj (x)|(| ln ρˆnj (x)| + 2 |w ˆjn (x)| + 2 |ˆ v,j (x)|)+ n |∇ϕˆnj (x)|(|∇ ln ρˆnj (x)| + 2 |∇w ˆjn (x)| + 2 |∇ˆ v,j (x)|)
Then, in view of (3.10), (3.11) and (3.16), we have sup (6.18)
ˆ n \U n x∈B j j
sup ˆ n \U n x∈B j j
n (| ln ρˆnj (x)| + 2 |w ˆjn (x)| + 2 |ˆ v,j (x)|) ≤ C| ln δ|
as δ → 0
n (|∇ ln ρˆnj (x)| + 2 |∇w ˆjn (x)| + 2 |∇ˆ v,j (x)|) ≤ Cδ
as δ → 0.
Hence, since supx∈Bˆ n \U n |∆ϕˆnj (x)| ≤ Cδ 2 and supx∈Bˆ n \U n |∇ϕˆnj (x)| ≤ Cδ, we may j j j j conclude X (6.19) k Cjn (x)kYˆδ ≤ Cδ 2 | ln δ|as δ → 0. n;j
Then, in view of (3.10),(3.11), (3.15) and lemma 6.2, the map F,δ : B1 ⊂ ˆ δ → Yˆδ is well defined and smooth provided > 0 is taken sufficiently small and H ˆ δ : kuk ˆ ≤ 1}. Moreover, in view of lemma 6.2, we have B1 = {u ∈ H Hδ Lemma 6.3. For any > 0 and δ > 0 sufficiently small, we have C as δ → 0. (6.20) kF,δ (0)kYˆδ ≤ 2 δ 2 |ln δ|, Proof. From the definition of the map F,δ we have 2 n n 1 kF,δ (0)kYˆδ ≤C 2 sup kϕˆnj ρˆnj e (wˆj +ˆv,j ) [Rjn (x) − 1]kYˆ n + j n;j + sup kϕˆnj (ˆ ρnj )2 e2
(6.21)
2
n (w ˆjn +ˆ v,j )
[(Rjn (x))2 − 1]kYˆ n + j
n;j
1 X n + 2k C (x)kYˆδ n;j j By lemma 6.2-(i), (3.10), (3.11) and (3.15) we have kϕˆnj ρˆnj e
2
n (w ˆjn +ˆ v,j )
[Rjn (x) − 1]kYˆ n ≤C j
(6.22)
sup ˆ n \U ˆn x∈B j j
≤Cδ 4−β()
(ˆ ρnj e
2
n (w ˆjn +ˆ v,j )
|Rjn (x) − 1|)
as δ → 0.
with β() → 0 as → 0. Analogously, we get also (6.23)
kϕˆnj (ˆ ρnj )2 e2
2
n (w ˆjn +ˆ v,j )
[(Rjn (x))2 − 1]kYˆ n ≤ o(δ 4−β() )
as δ → 0,
j
with β() → 0 as → 0. Therefore, in view of lemma 6.2-(ii) we may conclude that C (6.24) kF,δ (0)kYˆδ ≤ 2 (δ 2 | ln δ| + o(δ 2 )), as δ → 0.
16
M. NOLASCO
ˆ δ → Yˆδ given by Now, we consider the operator A,δ ≡ DF,δ (0) : H X 2 n n 2 n n A,δ ≡ DF,δ (0) = ∆ + ϕˆnj ρˆnj Rjn (x)e (wˆj +ˆv,j ) (1 − 22 ρˆnj Rjn (x)e (wˆj +ˆv,j ) ). n;j
We have ˆ δ we have Lemma 6.4. For any > 0 and δ > 0 sufficiently small, and any h ∈ H n n 4−β() n k(A,δ − Aˆj )ϕˆj hkYˆ n ≤ Cδ kϕˆj hkXˆ n , as δ → 0, j
j
with β() → 0 as → 0. Proof. From the definition of the operators A,δ and Aˆn,j we have k(A,δ − Aˆn,j )ϕˆnj hkYˆ n ≤ kˆ ρnj e
2
n (w ˆjn +ˆ v,j )
j
2
(ϕˆnj Rjn (x) − 1)ϕˆnj hkYˆ n + j
n
n
+2 k(ˆ ρnj )2 e2 (wˆj +ˆv,j ) (ϕˆnj (Rjn (x))2 − 1)ϕˆnj hkYˆ n + j X k k k 2 ( w ˆlk +ˆ v,l ) k n +k ϕˆl ρˆl e Rl (x)ϕˆj hkYˆ n +
(6.25)
j
(k,l)6=(n,j)
X
+2 k
ϕˆkl (ˆ ρkl )2 e2
2
k (w ˆlk +ˆ v,l )
(Rlk (x))2 ϕˆnj hkYˆ n . j
(k,l)6=(n,j)
Now, in view of lemma 6.2-(i), we have ke
2
n (w ˆjn +ˆ v,j ) n ρˆj (ϕˆnj Rjn (x)
≤
(6.26)
sup ˆ n \U n x∈B j j
− 1)ϕˆnj hkYˆ n ≤ j
2 n n (e (wˆj +ˆv,j ) ρˆnj |Rjn (x)
− 1|)kϕˆnj hkXˆ n ≤ j
≤Cδ 4−β() kϕˆnj hkXˆ n j
and, with an analogous estimate, it is easy to check that ke2
(6.27)
2
n (w ˆjn +ˆ v,j )
(ˆ ρnj )2 (ϕˆnj (Rjn (x))2 − 1)ϕˆnj hkYˆ n ≤ o(δ 4−β() )kϕˆnj hkXˆ n . j
j
Moreover, by (6.14), we have k
X
ϕˆkl ρˆkl Rlk (x)e
2
k (w ˆlk +ˆ v,l )
ϕˆnj hkYˆ n ≤ j
(k,l)6=(n,j)
≤C
(6.28)
sup
sup
ˆ n ∩B ˆk (k,l)6=(n,j) x∈B j
(ˆ ρkl Rlk (x)e
2
k (w ˆlk +ˆ v,l )
)kϕˆnj hkXˆ n ≤ j
l
≤Cδ 4−β() kϕˆnj hkXˆ n j
and, analogously, we have (6.29)
k
X
ϕˆkl (ˆ ρkl )2 e2
2
k (w ˆlk +ˆ v,l )
(Rlk (x))2 ϕˆnj hkYˆ n ≤ o(δ 4−β() )kϕˆnj hkXˆ n . j
j
(k,l)6=(n,j)
Consequently, we obtain (6.30)
k(A,δ − Aˆn,j )ϕˆnj hkYˆ n ≤ Cδ 4−β() kϕˆnj hkXˆ n , j
as δ → 0,
j
with β() → 0 as → 0. In view of lemma 6.4 and proposition 5.2 we have
NON-TOPOLOGICAL N-VORTEX CONDENSATES ...
17
Proposition 6.5. For any > 0 sufficiently small there exists δ > 0 such that for ˆ δ → Yˆδ is invertible. Moreover, any δ ∈ (0, δ ), the operator A,δ = DF,δ (0) : H −1 C kA,δ k ≤ 2 (uniformly w.r.t. δ ∈ (0, δ )). Proof. For any n = (n1 , n2 ) ∈ Z2 and j = 0, . . . , s, we introduce the smooth P ϕ ˆn (x) functions gˆjn (x) = P j k 2 gjn )2 (x) = 1, for any x ∈ R2 . 1 . Note that n;j (ˆ (
ˆi k;i (ϕ
) (x)) 2
ˆ δ given by Then, we consider the auxiliary linear operator S,δ : Yˆδ → H (6.31)
S,δ =
X
gˆjn (Aˆn,j )−1 gˆjn .
n;j
By proposition 5.2, we have that for any h ∈ Yˆδ
(6.32)
C kS,δ hkHˆ δ ≤ C sup kˆ gjn (Aˆn,j )−1 gˆjn hkXˆ n ≤ 2 khkYˆδ j n;j
ˆ δ is well defined and bounded hence, for any > 0 sufficiently small S,δ : Yˆδ → H (uniformly w.r.t. δ > 0). Now, we claim that for > 0 sufficiently small there exists δ such that for any ˆδ → H ˆ δ and A,δ S,δ : Yˆδ → Yˆδ are δ ∈ (0, δ ), both the operators S,δ A,δ : H invertible. Indeed, we have S,δ A,δ =IHˆ δ +
X
A,δ S,δ =IYˆδ +
X
gˆjn (Aˆn,j )−1 (A,δ − Aˆn,j )ˆ gjn +
X
− Aˆn,j )(Aˆn,j )−1 gˆjn +
X
n;j
(6.33)
gˆjn (Aˆn,j )−1 [A,δ , gˆjn ]
n;j
gˆjn (A,δ
n;j
ˆ[A,δ , gˆn ](Aˆn )−1 g n . j ,j j
n;j
Hence it is enough to prove that the last two terms in (6.33) are sufficiently small (in the norm operator) for δ > 0 sufficiently small. Indeed, by lemma 6.4, for any ˆ δ we have h∈H k
X
gˆjn (Aˆn,j )−1 (A,δ − Aˆn,j )ˆ gjn hkHˆ δ ≤C sup kˆ gjn (Aˆn,j )−1 (A,δ − Aˆn,j )ˆ gjn hkXˆ n ≤ j
n;j
n;j
C sup k(A,δ − Aˆn,j )ϕˆnj hkYˆ n ≤ j 2 n;j C ≤ 2 δ 4−β() sup kϕˆnj hkXˆ n j n;j ≤
with β() → 0 as → 0. Moreover, we have sup kϕˆnj gˆjn (Aˆn,j )−1 [A,δ , gˆjn ]hkXˆ n ≤ j
n;j
C ≤ 2 sup n;j
C sup k[∆, gˆjn ]hkYˆ n ≤ j 2 n;j ! 21
Z (1 + (δ|x − ˆ n \U n B j j
pˆnj |)2+α )(∇ˆ gjn ∇h
+
(∆ˆ gjn )h)2 δ 2
dx
.
18
M. NOLASCO
Now, recalling that supx∈Bˆ n |∆ˆ gjn (x)| ≤ Cδ 2 , supx∈Bˆ n |∇ˆ gjn (x)| ≤ Cδ and that j
P
k;i
j
ϕˆki (x) = 1, for any x ∈ R2 , we have Z 12 C 2 2 |∇h| δ dx + ≤ 2δ R2 12 Z 2 X h C ϕˆki δ 2 dx ≤ + 2 δ 2 sup n n (1 + (δ|x − pˆnj |)2+α ) ˆ n;j Bj \Uj
sup kϕˆnj gˆjn (Aˆn,j )−1 [A,δ , gˆjn ]hkXˆ n j n;j
k,i
C ≤ 2δ
12 C |∇h| δ dx + 2 δ 2 sup kϕˆnj hkXˆ n . j n;j R2
Z
2 2
Then, by standard density argument, integration by parts and H¨ older inequality we have 12 Z Z X 2 h |∇h|2 δ 2 dx ≤ ϕˆki δ 2 dx × (1 + (δ|x − pˆnj |)2+α ) R2 R2 k,i (6.34) 12 Z X ϕˆki |∆h|2 (1 + (δ|x − pˆnj |)2+α )δ 2 dx . × R2 k,i
Consequently, we may conclude (6.35)
sup kϕˆnj gˆjn (Aˆn,j )−1 [A,δ , gˆjn ]hkXˆ n ≤ n;j
j
C (δ + o(δ)) sup kϕˆnj hkXˆ n . j 2 n;j
Hence, kS,δ A,δ − IHˆ δ k → 0 as δ → 0. With analogous estimates we also get kA,δ S,δ − IYˆδ k → 0 as δ → 0. Therefore, for any > 0 sufficiently small there exists δ > 0 such that for any δ ∈ (0, δ ˆδ → H ˆ δ and A,δ S,δ : Yˆδ → Yˆδ are invertible and both the operators S,δ A,δ : H the inverse operators are uniformly bounded (w.r.t. > 0 and δ > 0 sufficiently ˆ δ → Yˆδ small), namely k(A,δ S,δ )−1 k ≤ 2 and k(S,δ A,δ )−1 k ≤ 2. Then, A,δ : H −1 −1 is invertible with inverse given by (A,δ ) = S,δ (A,δ S,δ ) = (S,δ A,δ )−1 S,δ : ˆ δ . To conclude the proof we note that for any h ∈ Yˆδ we have Yˆδ → H (6.36)
k(A,δ )−1 hkHˆ δ = k(S,δ A,δ )−1 S,δ hkHˆ δ ≤ CkS,δ hkHˆ δ ≤
C khkYˆδ . 2
ˆ δ : kuk ˆ ≤ r}. For any r > 0 we set Br = {u ∈ H Hδ Now, for any > 0 sufficiently small δ >∈ (0, δ ), with δ > 0 given by proposiˆ δ ) defined by tion 6.5, we introduce the nonlinear map G,δ ∈ C 1 (B1 , H (6.37)
G,δ (z) = z − A−1 ,δ F,δ (z).
Then, the fixed points of G,δ are solutions of the functional equation F,δ (z) = 0. Theorem 6.6. For any > 0 sufficiently small there exists δ > 0, δ → 0 as → 0, such that for any δ ∈ (0, δ ) there exists Rδ = O(δ γ ) > 0, for some γ ∈ (1, 2), and ∗ of G,δ in BRδ . a unique fixed point z,δ Proof. The proof is a direct application of the Banach fixed point theorem.
NON-TOPOLOGICAL N-VORTEX CONDENSATES ...
19
First, note that DG,δ (0) = 0. Then, by proposition 6.5, for any z ∈ B1 we have kDG,δ (z)k =kDG,δ (z) − DG,δ (0)k ≤ kA−1 ,δ kkDF,δ (z) − A,δ k ≤ (6.38) ≤
C kDF,δ (z) − A,δ k 2
ˆδ, and, for any h ∈ H k(DF,δ (z) − A,δ )hkYˆδ ≤ C sup kϕˆnj (DF,δ (z) − A,δ )hkYˆ n ≤ j
n;j
≤C
(6.39)
2 n n 2 sup kϕˆnj e (wˆj +ˆv,j ) ρˆnj Rjn (x)(e z n;j
+C2 sup kϕˆnj e2
2
n ) (w ˆjn +ˆ v,j
n;j
−
1)ϕˆnj hkYˆ n + j
(ˆ ρnj )2 (Rjn (x))2 (e2
2
z
− 1)ϕˆnj hkYˆ n . j
Now, as pointed out in remark 6.1, we have sup 2 ρˆnj e
(6.40)
2
n (w ˆjn +ˆ v,j )
x∈R2
Rjn (x) ≤ 1,
and noting that (6.41)
|e
2
z(x)
− 1| ≤ 2 |z(x)|
Z
1
e
2
tz(x)
dt ≤ 2 kzk∞ e
2
kzk∞
,
∀x ∈ R2 ,
0
by Sobolev inequality and in view of remark 5.1, we may conclude that C k(DF,δ (z) − A,δ )hkYˆδ ≤ 2 2 2 C ≤ 2 sup kϕˆnj zk∞ e kzk∞ kϕˆnj hkXˆ n (1 + e kzk∞ ) ≤ j n;j C C ≤ 2 sup kϕˆnj zkXˆ n (1 + sup kϕˆnj zkXˆ n )kϕˆnj hkXˆ n ≤ j j j δ n;j δ n;j C C ≤ 2 kzkHˆ δ (1 + kzkHˆ δ )khkHˆ δ . δ δ Consequently, setting Rδ = δ γ r¯ for some γ ∈ (1, 2) and a suitable r¯ > 0 (independent of > 0 and δ > 0), we may conclude that for any > 0 sufficiently small there exists δ > 0 such that for any δ ∈ (0, δ ), we have kDG,δ (z)hkYˆδ ≤
(6.42)
kDG,δ (z)k ≤
1 , 2
∀z ∈ BRδ .
Now, for any z ∈ BRδ , we have kG,δ (z)kHˆ δ ≤ kG,δ (z) − G,δ (0)kHˆ δ + kG,δ (0)kHˆ δ ≤ 1 ≤ kzkHˆ δ + kA−1 ˆδ , F,δ (0)kH 2 then, by proposition 6.5 and lemma 6.3, we obtain C 2 −1 (6.44) kA−1 ˆ δ ≤ kA,δ kkF,δ (0)kYˆδ ≤ 2 δ | ln δ| ,δ F,δ (0)kH Therefore, for any > 0 sufficiently small there exists δ > 0 such that for any 1 δ ∈ (0, δ ) we have kA−1 ˆ δ ≤ 2 Rδ . Therefore we can conclude that the map ,δ F,δ (0)kH G,δ is a (strict) contraction in BRδ and that G,δ (BRδ ) ⊂ BRδ , for any δ ∈ (0, δ ). By the Banach fixed-point theorem, for any > 0 sufficiently small and δ ∈ (0, δ ), ∗ ∗ there exists unique z,δ ∈ BRδ , such that F,δ (z,δ ) = 0. (6.43)
Remark 6.7. In view of the estimates in proposition 6.5 and theorem 6.6 we have 2 that δ = o( γ−1 , with γ ∈ (1, 2) as in theorem 6.6.
20
M. NOLASCO
Thus, in view of theorem 6.6, we may conclude that the the function (6.45)
u ˆ,δ (x) =
s XX n∈Z2
n ∗ ϕˆnj (x)(ln ρˆnj (x) + 2 w ˆjn (x) + 2 vˆ,j (x)) + 2 z,δ (x)
j=0
is a solution of the equation (2.4). Now, we want to prove that u ˆ,δ (x) is in fact a periodic solution of (2.4). We have ∗ Proposition 6.8. For any > 0 sufficiently small and any δ ∈ (0, δ ) let z,δ (x) be the unique fixed point of G,δ in BRδ . Then u ˆ,δ (x + eˆk ) = u ˆ,δ (x), for any x ∈ R2 and for k = 1, 2.
Proof. Note that u ˆ,δ (x + eˆk ) =
∗ 2 z,δ (x
+ eˆk ) +
s XX
n ϕˆnj (x)(ln ρˆnj (x)) + 2 w ˆjn (x) + 2 vˆ,j (x))
n∈Z2 j=0 ∗ ∗ Hence it is sufficient to prove that z,δ (x + eˆk ) = z,δ (x). ∗ First, we claim that z,δ ( · + eˆk ) ∈ BRδ . Indeed, 0
∗ ∗ kϕˆnj z,δ ( · + eˆk )kXˆ n = kϕˆnj z,δ kXˆ n0
(6.46)
j
0
(n01 , n02 ) 2
j
n0i
2
with n = ∈ Z and = ni + δik . Hence, taking the supremum over the index in Z and over j = 0, . . . , s, we may conclude ∗ ∗ kz,δ ( · + eˆk )kHˆ δ = kz,δ kHˆ δ ≤ Rδ .
(6.47)
∗ ∗ Moreover, it is easy to check that if F,δ (z,δ ) = 0 we have also F,δ (z,δ ( · + eˆk )) = 0 ∗ hence z,δ ( · + eˆk ) is also a fixed point of G,δ in BRδ and by uniqueness, we may ∗ ∗ conclude z,δ (x + eˆk ) = z,δ (x) for any x ∈ R2 and for k = 1, 2.
7. Asymptotic properties of the N-vortex solutions In the previous section we prove the existence of a periodic N-vortex solution of (2.3). More precisely, for any k = 2δ > 0 with > 0 sufficiently small and δ ∈ (0, δ ), with δ given by theorem 6.6 (see remark 6.7), we get a solution of equation (2.3) u,δ (x) = u ˆ,δ ( xδ ) + ln 2 given by u,δ (x) = ln 2 +
(7.1)
= ln 2 + + 2
s X
1 x ∗ ϕj (x)u∗,mj ( |x − pj |)) + 2 z,δ ( )= δ δ j=0
s X
1 ϕj (x) ln ρmj ( |x − pj |)+ δ j=0
s X
1 1 x ∗ ∗ ϕj (x)(wmj ( |x − pj |) + v,m ( |x − pj |)) + 2 z,δ ( ) j δ δ δ j=0
for any x ∈ Ω, satisfying periodic boundary conditions on ∂Ω, and where u∗,mj ( 1δ |x− pj |)) is the single vortex solutions radially symmetric w.r.t. the vortex point pj , and multiplicity mj . Now, for k = 2δ, we study the asymptotic properties of |φk | = exp( 12 u,δ ) as → 0 and δ ∈ (0, δ ), with δ → 0 as → 0. We have the following result:
NON-TOPOLOGICAL N-VORTEX CONDENSATES ...
21
Proposition 7.1. Let k = 2δ > 0 with > 0 sufficiently small and δ ∈ (0, δ ), with δ → 0 as → 0. Then, |φk | = exp( 12 u,δ ), with u,δ given by (7.1) satisfies the following properties: (i) |φk | < 1 on Ω and vanishes exactly at pj with multiplicity mj , (j = 1, . . . , s); q (ii) |φk | → 0 in Cloc (Ω \ {p1 , . . . , ps }) as k → 0+ , for any q ≥ 0; Ps (iii) F1,2 (φk ) → 2π j=1 mj δpj in the sense of measure, as k → 0+ .
Proof. (i) Since u,δ is a solution of equation (2.3), |φk | = exp( 21 u,δ ) < 1 follows immediately from the maximum principle. 8(mj +1)2 |x|2mj ∗ ∗ Moreover, since ρmj ( 1δ |x − pj |) = (1+|x| and wmj , v,m and z,δ are 2mj +2 2 j ) continuous function, we have |φk (x)| = Πj |x − pj |mj f (x), with f (x) a continuous positive function. (ii) Let K be a compact subset of Ω \ {p1 . . . ps }. In view of (3.10), (3.11), (3.15), theorem 6.6 and remark 5.1 , we have 1 ρmj ( |x − pj |) ≤ Cδ 4+2mj as δ → 0 δ x∈K∩Bj 1 sup |wmj ( |x − pj |)| ≤ C| ln δ| as δ → 0 δ x∈K∩Bj 1 ∗ ( |x − pj |)| ≤ o()| ln δ| as δ → 0 sup |v,m j δ x∈K∩Bj sup
(7.2)
∗
kz,δ kXˆ δ x Rδ ∗ ≤ Cδ γ−1 sup |z,δ ( )| ≤ C ≤ δ δ δ x∈Ω
as δ → 0.
Therefore, for k = k() = 2δ with > 0 sufficiently small and δ ∈ (0, δ ), with δ → 0 as → 0 (see remark 6.7), we have that (7.3)
sup |φk (x)|2 ≤ C sup x∈K
j
sup
ϕj (x)|φk (x)|2 ≤ C2 δ 4+2m−β()
x∈K∩Bj
with m = inf j {mj } and β() → 0, as → 0. Thus, we may conclude that for any compact set K ⊂ Ω \ {p1 . . . ps } (7.4)
sup |φk (x)| → 0
as → 0.
x∈K
In fact, since by elliptic regularity u,δ ∈ C ∞ (Ω \ {p1 . . . ps }), we have that (7.5)
1 |φk | = exp( u,δ ) → 0 2
q in Cloc (Ω \ {p1 . . . ps }) ∀q ≥ 0
as → 0. (iii) By the periodic conditions on ∂Ω, integrating the equation (2.2) over Ω, we get Z 1 (7.6) |φk |2 (1 − |φk |2 ) = πN. k2 Ω Now, choosing any r > 0 such that Br (pj ) ∩ Br (pi ) = ∅ for i 6= j ∈ {1, . . . , s}, in view of (7.3), for k = k() = 2δ with > 0 sufficiently small and δ ∈ (0, δ ), with δ → 0 as → 0 (see remark 6.7), we have Z 1 (7.7) |φk |2 (1 − |φk |2 ) ≤ Cδ 2+2mj −β() → 0 ∀j = 1, . . . , s k 2 Ω\Br (pj )
22
M. NOLASCO
as δ → 0. Therefore, for k = k() = 2δ with > 0 sufficiently small and δ ∈ (0, δ ), we may conclude that s X 2 (7.8) F1,2 (φk ) = 2 |φk |2 (1 − |φk |2 ) → 2π mj δpj k j=1 in the sense of measure, as → 0. References [1] S. Angenent, The Shadowing Lemma for Elliptic PDE , Dynamics of Infinite Dimensional Systems, (S.N. Chow and J.K. Hale eds.) F37 (1987). [2] L. Caffarelli and Y. Yang, Vortex condensation in the Chern-Simons-Higgs model: an existence theorem, Comm. Math. Phys. 168 (1995), 321–366. [3] D. Chae and O. Imanuvilov, The existence of non-topological multivortex solutions in the relativistic selfdual Chern-Simons theory, Comm. Math. Phys. 215 (2000), 119–142. [4] D. Chae and N. Kim, Topological multivortex solutions of the selfdual Maxwell Chern-Simons system, J. Diff. Eq. 134 (1997), 154–182. [5] W. Ding, J. Jost, J. Li, X. Peng, and G. Wang, Self duality equations for Ginzburg-Landau and Seiberg-Witten type functionals with 6th order potential, Comm. Math. Phys. 217(2001), 383–407. [6] W. Ding, J. Jost, J. Li, and G. Wang, An analysis of the two-vortex case in the ChernSimons-Higgs model, Calc. Var. and PDE 7 (1998), 87–97. [7] W. Ding, J. Jost, J. Li, and G. Wang, Multiplicity results for the two-vortex Chern-Simons Higgs model on the two-sphere, Comment. Math. Helv. 74 (1999), no. 1, 118–142. [8] G. Dunne, Selfdual Chern-Simons theories, Lectures Notes in Physics, vol. 36, SpringerVerlag, Berlin, New York, 1995. [9] J. Hong, Y. Kim, and P. Pac, Multivortex solutions of the Abelian Chern-Simons theory, Phys. Rev. Lett. 64 (1990), 2230–2233. [10] R. Jackiw and E. Weinberg, Selfdual Chern-Simons vortices, Phys. Rev. Lett. 64 (1990), 2234–2237. [11] M. Nolasco and G. Tarantello, Double vortex condensates in the Chern-Simons-Higgs theory, Calc. Var. and PDE. 9 (1999), 31–94. [12] M. Nolasco and G. Tarantello, Vortex condensates for the SU(3) Chern-Simons theory, Comm. Math. Phys. 213 (2000), 599–639. [13] T. Ricciardi and G. Tarantello, Self-dual vortices in the Maxwell-Chern-Simons-Higgs theory, Comm. Pure and Appl. Math. 53 (2000), no. 7, 811–851. [14] J. Spruck and Y. Yang, The existence of non-topological solutions in the self-dual ChernSimons theory, Comm. Math. Phys. 149 (1992), 361–376. [15] J. Spruck and Y. Yang, Topological solutions in the self-dual Chern-Simons theory: existence and approximation, Ann. I.H.P. Analyse Nonlin. 12 (1995), 75–97. [16] G. ’t Hooft, A property of electric and magnetic flux in nonabelian gauge theories, Nucl. Phys. B 153 (1979), 141–160. [17] G. Tarantello, Multiple condensates solutions for the Chern-Simons-Higgs theory, J. Math. Phys. 37 (1996), 3769–3796. [18] C. Taubes, Arbitrary n-vortex solutions to the first order Ginzburg-Landau equations, Comm. Math. Phys. 72 (1980), 277–292. [19] R. Wang, The existence of Chern-Simons vortices, Comm. Math. Phys. 137 (1991), 587–597. [20] R. Wang and L. Zhang, Non-Topological solutions of relativistic SU(3) Chern-Simons-Higgs model, Comm. Math. Phys. 202 (1999), 501–515. [21] Y. Yang, The Relativistic Non-Abelian Chern-Simons Equations, Comm. Math. Phys. 186 (1997), 199–218. ` di L’Aquila, Via Vetoio, Cop(M. Nolasco) Dipartimento di Matematica, Universita pito, 67010 L’Aquila, Italy E-mail address: [email protected]
