Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions

DOI:10.1051/cocv:2008055 www.esaim-cocv.org

MULTI-PEAK SOLUTIONS FOR MAGNETIC NLS EQUATIONS WITHOUT NON-DEGENERACY CONDITIONS

Silvia Cingolani

, Louis Jeanjean

and Simone Secchi

Abstract. In this work we consider the magnetic NLS equation

i∇ −A(x) ₂

u+V(x)u−f(|u|²)u = 0 inR^N (0.1)

whereN ≥3,A:R^N→R^N is a magnetic potential, possibly unbounded,V:R^N →Ris a multi-well electric potential, which can vanish somewhere,fis a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solutionu:R^N→Cto (0.1), under conditions on the nonlinearity which are nearly optimal.

Mathematics Subject Classification.35J20, 35J60.

Received October 16, 2007. Revised March 13, 2008.

Published online August 20, 2008.

1. Introduction

We study the existence of a standing wave solution ψ(x, t) = exp(−iEt/)u(x),E ∈R, u:R^N →Cto the time-dependent nonlinear Schr¨odinger equation in the presence of an external electromagnetic field

i∂ψ

∂t =

i∇ −A(x) ₂

ψ+V(x)ψ−f(|ψ|²)ψ, (t, x)∈R×R^N. (1.1) Hereis the Planck’s constant,ithe imaginary unit,A:R^N →R^N denotes a magnetic potential andV:R^N → Ran electric potential. This leads us to solve the complex semilinear elliptic equation

i∇ −A(x) ₂

u+ (V(x)−E)u−f(|u|²)u = 0, x∈R^N. (1.2)

Keywords and phrases. Nonlinear Schr¨odinger equations, magnetic fields, multi-peaks.

∗ The first author is supported by MIUR, national project Variational and topological methods in the study of nonlinear phenomena (PRIN 2005).

∗∗ The third author is supported by MIUR, national project Variational methods and nonlinear differential equations.

1 Dipartimento di Matematica, Politecnico di Bari, via Orabona 4, 70125 Bari, Italy.s.cingolani@poliba.it

2Equipe de Math´´ ematiques (UMR CNRS 6623), 16 Route de Gray, 25030 Besan¸con, France. louis.jeanjean@univ-fcomte.fr

3 Dipartimento di Matematica ed Applicazioni, Universit`a di Milano-Bicocca, via Cozzi 53, 20125 Milano, Italy.

Simone.Secchi@unimib.it

Article published by EDP Sciences c EDP Sciences, SMAI 2008

In the work we are interested to seek for solutions of (1.2), which exist for small value of the Planck constant >0. From a mathematical point of view, the transition from quantum to classical mechanics can be formally performed by letting →0, and such solutions, which are usually referredsemiclassical bound states, have an important physical meaning.

For simplicity and without loss of generality, we set=εand we shiftE to 0. Setv(x) =u(εx), A_ε(x) = A(εx) andV_ε(x) =V(εx),equation (1.2) is equivalent to

1

i∇ −A_ε(x) ₂

v+V_ε(x)v−f(|v|²)v= 0, x∈R^N. (1.3) In recent years a considerable amount of work has been devoted to investigating standing wave solutions of (1.1) in the case A = 0. Among others we refer to [1,2,7,9–11,13,17,18,24–26,28,30,34,36,38,40,45]. On the contrary still relatively few papers deal with the case A = 0, namely when a magnetic field is present.

The first result on magnetic NLS equations is due to Esteban and Lions. In [27], they prove the existence of standing waves to (1.1) by a constrained minimization approach, in the case V(x) = 1, for > 0 fixed and for special classes of magnetic fields. Successively in [35], Kurata showed that equation (1.3) admits, under some assumptions linking the magnetic and electric potentials, a least energy solution and that this solution concentrates near the set of global minima of V, as → 0. It is also proved that the magnetic potential A only contributes to the phase factor of the solution of (1.3) for>0 sufficiently small. A multiplicity result for solutions of (1.3) near global minima ofV has been obtained in [16] using topological arguments. A solution that concentrates as→0 around an arbitrary non-degenerate critical point ofV has been obtained in [19] but only for bounded magnetic potentials. Subsequently this result was extended in [20] to cover also degenerate, but topologically non trivial, critical points ofV and to handle general unbounded magnetic potentialsA. IfA and V are periodic functions, the existence of various type of solutions for >0 fixed has been proved in [3]

by applying minimax arguments. We also mention the works [4,15] that deal with critical nonlinearities.

Concerning multi-well electric potentials, an existence result of multi-peak solutions to the magnetic NLS equation (1.3) is established by Bartschet al. in [5], assuming that the functionf is increasing on (0,+∞) and satisfies the Ambrosetti-Rabinowitz’s superquadraticity condition. Also, in [5], an isolatedness condition on the least energy level of the limiting equation

−Δu+bu−f(|u|²)u= 0, u∈H¹(R^N,C) is required to hold for anyb >0.

In the present paper we prove an existence result of multi-peak solutions to (1.3), under conditions on f, that we believe to be nearly optimal. In particular we drop the isolatedness condition, required in [5] and we cover the case of nonlinearities, which are not monotone.

Precisely, the following conditions will be retained.

(A1): A:R^N →R^N is of classC¹.

(V1): V ∈C⁰(R^N,R), 0≤V₀= inf_x∈RNV(x) and lim inf_|x|→∞V(x)>0.

(V2): There are bounded disjoint open setsO¹, . . . , O^k such that 0< m_i= inf

x∈OⁱV(x)< min

x∈∂OⁱV(x) fori= 1, . . . , k.

For eachi∈ {1, . . . , k},we define

Mⁱ={x∈Oⁱ |V(x) =m_i} and we setZ ={x∈R^N |V(x) = 0} andm= min

i∈{1,...,k}m_i.

On the nonlinearityf, we require that (f0): f: (0,+∞)→Ris continuous;

(f1): lim

t→0⁺f(t) = 0 ifZ =∅, and lim sup

t→0⁺ f(t²)/t^μ<+∞for someμ >0 if Z =∅;

(f2): there exists some 0< p < _N⁴₋₂, N≥3 such that lim sup_t→+∞ f(t²)/t^p <+∞;

(f3): there existsT >0 such that ¹₂mTˆ ²< F(T²),where

F(t) = _t

0 f(s)ds, mˆ = max

i∈{1,...,k}m_i. Now by assumption(V1), we can fixm > 0 such that

m <min

m, lim inf

|x|→∞V(x)

(1.4) and define ˜V_ε(x) = max{m, V _ε(x)}.LetH_εbe the Hilbert space defined by the completion ofC₀^∞(R^N,C) under the scalar product

u, vε= Re

R^N

1

i∇u−A_ε(x)u 1

i∇v−A_ε(x)v

+ ˜V_ε(x)uv dx (1.5)

and · εthe associated norm.

In the present work, we shall prove the following main theorem.

Theorem 1.1. LetN ≥3. Suppose that (A),(V1)–(V2)and (f0)–(f3)hold. Then for anyε >0sufficiently small, there exists a solution u_ε∈H_εof (1.3)such that |uε|has k local maximum pointsxⁱ_ε∈Oⁱ satisfying

ε→0lim max

i=1,...,kdist(εxⁱ_ε,Mⁱ) = 0, and for which

|uε(x)| ≤C₁exp

−C₂ min

i=1,...,k|x−xⁱ_ε|

for some positive constants C₁, C₂. Moreover for any sequence (ε_n) ⊂ (0, ε] with ε_n → 0 there exists a subsequence, still denoted (ε_n), such that for each i ∈ {1, . . . , k} there exist xⁱ ∈ Mⁱ with ε_nxⁱ_εn → xⁱ, a constant w_i ∈RandU_i∈H¹(R^N,R)a positive least energy solution of

−ΔU_i+m_iU_i−f(|U_i|²)U_i= 0, U_i∈H¹(R^N,R); (1.6) for which one has

u_εn(x) = k i=1

U_i x−xⁱ_εn

eⁱ(wi+A(xⁱ)(x−xⁱ_εn)) +K_n(x) (1.7) whereK_n∈H_εn satisfiesK_nHεn =o(1)asε_n→0.

Remark 1.2. Arguing as in [21], we can develop a bootstrap argument, and prove that the solutionu_ε∈H_ε, found in Theorem1.1, belongs toC¹(R^N,C). Indeed, setu_ε=v+iw, with v, wreal valued, we have

−Δv+V_εv=G:=f(|uε|²)v−2A_ε· ∇w− |Aε|²v+ (divA_ε)w and

−Δw+V_εw=H:=f(|uε|²)w+ 2A_ε· ∇v− |Aε|²w+ (divA_ε)v.

Since u_ε∈ H_ε, it follows that for each K bounded set inR^N, u_ε ∈H¹(K,C). Thereforev, w ∈ H¹(K,R)⊂ L^2∗(K,R) and by (f2), G, H ∈ L^s(K,R), where s = min{2^∗/(p−1),2}. Standard regularity theory implies thatv, w∈W^2,s(K). If 2s < N we can argue as before and derive thatv, w∈L^{N s/(N−2s)}(K,R) and∇v,∇w∈ L^{N s/(N−s)}(K,R). After a finite number of steps, we have thatv, w∈W^2,q(K) for anyq∈[1,+∞[ and by the Sobolev embedding theorems,v, w∈C^1,α(K,R), with 0< α <1.

Remark 1.3. If we assume the uniqueness of the positive least energy solutions of (1.6) it is not necessary to pass to subsequences to get the decomposition (1.7) in Theorem1.1.

The proof of Theorem1.1follows the approach which is developed in [9] to obtain multi-peak solutions when A= 0.Roughly speaking we search directly for a solution of (1.3) which consists essentially ofkdisjoints parts, each part being close to a least energy solution of (1.6) associated to the corresponding Mⁱ. Namely in our approach we take into account the shape and location of the solutions we expect to find. Thus on one hand we benefit from the advantage of the Lyapunov-Schmidt reduction type approach, which is to discover the solution around a small neighborhood of a well chosen first approximation. On the other hand our approach, which is purely variational, does not require any uniqueness nor non-degeneracy conditions.

We remark that differently from [9], we need to overcome many additional difficulties which arise for the presence of the magnetic potential. Indeed it is well known that, in general, there is no relationship between the spacesH_ε and H¹(R^N,C), namelyH_ε ⊂H¹(R^N,C) nor H¹(R^N,C) ⊂H_ε (see [27]). This fact explains, for example, the need to restrict to bounded magnetic potentials A when one uses a perturbative approach (see [19]). Our Lemma2.1and Corollary2.2give some insights of the relationship betweenH_ε andH¹(R^N,C) which proves useful in the proof of Theorem 1.1. We also answer positively a question raised by Kurata [35], regarding the equality between the least energy levels for the solutions of

−ΔU+bU=f(|U|²)U

whenU are sought inH¹(R^N,C) andH¹(R^N,R) respectively. See Lemma2.3for the precise statement.

In contrast to [5] we do not treat here the cases N = 1 and N = 2. For such dimensions applying the approach of [9] is more complex. It can be done when A= 0 and for the case of a single peak (see [12]) but it is an open question if Theorem1.1still holds when N= 1,2.

The work is organized as follows. In Section 2 we indicate the variational setting and prove some preliminary results. The proof of Theorem1.1 is derived in Section 3.

2. Variational setting and preliminary results

For any setB ⊂R^N andε >0, let B_ε={x∈R^N |εx∈B}.

Lemma 2.1. Let K ⊂ R^N be an arbitrary fixed bounded domain. Assume that A is bounded on K and 0< α≤V ≤β onK for someα, β >0. Then, for any fixedε∈[0,1], the norm

u²_Kε=

Kε

1

i∇ −A_ε(y)

u

²+V_ε(y)|u|²dy

is equivalent to the usual norm onH¹(K_ε,C). Moreover these equivalences are uniform,i.e. there existc₁, c₂>0 independent of ε∈[0,1]such that

c₁uKε≤ u_H¹_(Kε,C)≤c₂uKε. Proof. Our proof is inspired by the one of Lemma 2.3 in [3]. We have

Kε

|Aε(y)u|²dy≤ A_L^∞_(K)

Kε

|u|²dy

and

Kε

V_ε(y)|u|²dy≤ VL^∞(K)

Kε

|u|²dy.

Hence

Kε

1

i∇u−A_ε(y)u

²+V_ε(y)|u|²dy ≤

Kε

2 |∇u|²+|A_ε(y)u|²

+V_ε(y)|u|²dy

≤ 2

Kε

|∇u|²dy+ 2A_L^∞_(K)+V_L^∞_(K)

Kε

|u|²dy.

To prove the other inequality note that

Kε

1

i∇u−A_ε(y)u

²+V_ε(y)|u|²dy≥

Kε

|∇u|²− |A_ε(y)u|²+V_ε(y)|u|²dy.

We shall prove that, for somed >0 independent ofε∈[0,1],

Kε

|∇u| − |A_ε(y)u|²+V_ε(y)|u|²dy≥d

Kε

|∇u|²+|u|²dy. (2.1)

Arguing by contradiction we assume that there exist sequences (ε_n) ⊂ [0,1] and (u_εn) ⊂ H¹(K_εn,C) with uεn_H¹_(Kεn,C)= 1 such that

Kεn

|∇uεn| − |Aεn(y)u_εn|²+V_εn(y)|uεn|²dy < 1

n· (2.2)

Clearly (u_εn)⊂H¹(R^N,C) andu_εnH¹(R^N,C)= 1. Passing to a subsequence, u_εn u weakly inH¹(R^N,C).

SinceV_εn ≥α >0 onK_εn we see from (2.2) that necessarily

Kεn

|uεn|²dy→0.

Thusu_εn→0 inL²(R^N,C) strongly and in particularu_εn0 inH¹(R^N,C). Now

K_εn

|∇uεn| − |Aεn(y)u_εn|²dy=

K_εn

|∇uεn|²−2|Aεn(y)u_εn| |∇uεn|+|Aεn(y)u_εn|²dy

with

K_εn

|Aεn(y)u_εn| |∇uεn|dy→0.

Indeed we have

K_εn

|Aεn(y)u_εn| |∇uεn|dy ≤

K_εn

|Aεn(y)u_εn|²dy ¹₂

K_εn

|∇uεn|²dy ¹₂

≤ A_L^∞_(K)

Kεn

|uεn|²dy ¹₂

.

Thus

0 = lim sup

n→+∞

K_εn

|∇uεn| − |Aεn(y)u_εn|²dy≥lim sup

n→+∞

K_εn

|∇uεn|²dy.

But this is impossible since otherwise we would haveu_εn→0 strongly inH¹(R^N,C).

From Lemma2.1we immediately deduce the following corollary.

Corollary 2.2. Retain the setting of Lemma2.1.

(i) If K is compact, for anyε∈(0,1]the norm u²_K:=

K

1

i∇ −A_ε(y)

u

²+V_ε(y)|u|²dy is uniformly equivalent to the usual norm on H¹(K,C).

(ii) ForA₀∈R^N andb >0 fixed, the norm u²:=

R^N

1 i∇ −A₀

u

²+b|u|²dy is equivalent to the usual norm on H¹(R^N,C).

(iii) If (u_εn)⊂H¹(R^N,C) satisfiesu_εn = 0 on R^N \K_εn for anyn∈N and u_εn →u in H¹(R^N,C) then uεn−uεn→0 asn→ ∞.

Proof. Indeed (i) is trivial, to see (ii) just putε= 0 in Lemma2.1. Now (iii) follows from the uniformity of the

equivalence derived in Lemma2.1.

For future reference we recall the following Diamagnetic inequality: For everyu∈H_ε,

∇ i −A_ε

u

≥∇|u|, a.e. inR^N. (2.3)

See [27] for a proof. As a consequence of (2.3),|u| ∈H¹(R^N,R) for anyu∈H_ε. Now we define

M= k i=1

Mⁱ, O= k i=1

Oⁱ

and for any setB⊂R^N andα >0, B^δ ={x∈R^N |dist(x, B)≤δ}. Foru∈H_ε, let Fε(u) = 1

2

R^N|D^εu|²+V_ε|u|²dy−

R^NF(|u|²)dy (2.4)

where we setD^ε= (^∇_i −A_ε).Define

χ_ε(y) =

0 ify ∈O_ε,

ε^−6/μ ify /∈O_ε, χⁱ_ε(y) =

0 ify∈(Oⁱ)_ε, ε^−6/μ ify /∈(Oⁱ)_ε, and

Q_ε(u) =

R^Nχ_ε|u|²dy−1 ^p+2₂

+

, Qⁱ_ε(u) =

R^Nχⁱ_ε|u|²dy−1 ^p+2₂

+

. (2.5)

The functionalQ_εwill act as a penalization to force the concentration phenomena to occur insideO. This type of penalization was first introduced in [11]. Finally we define the functionals Γ_ε,Γ¹_ε, . . . ,Γ^k_ε :H_ε→Rby

Γ_ε(u) =Fε(u) +Q_ε(u), Γⁱ_ε(u) =Fε(u) +Qⁱ_ε(u), i= 1, . . . , k. (2.6) It is easy to check, under our assumptions, and using the diamagnetic inequality (2.3), that the function- als Γ_ε,Γⁱ_ε∈C¹(H_ε).So a critical point ofFεcorresponds to a solution of (1.3). To find solutions of (1.3) which concentrateinO asε→0,we shall look for a critical point of Γ_εfor whichQ_εis zero.

Let us consider fora >0 the scalar limiting equation of (1.3)

−Δu+au=f(|u|²)u, u∈H¹(R^N,R). (2.7) Solutions of (2.7) correspond to critical points of the limiting functionalL_a: H¹(R^N,R)→Rdefined by

L_a(u) =1 2

R^N |∇u|²+a|u|² dy−

R^NF(|u|²)dy. (2.8)

In [6], Berestycki and Lions proved that, for anya >0, under the assumptions(f0)–(f2)and(f3)with ˆm=a, there exists a least energy solution and that each solutionU of (2.7) satisfies the Pohozaev’s identity

N−2 2

R^N|∇U|²dy+N

R^Na|U|²

2 −F(|U|²)dy= 0. (2.9)

From this we immediately deduce that, for any solutionU of (2.7), 1

N

R^N|∇U|²dy=L_a(U). (2.10)

We also consider the complex valued equation, fora >0,

−Δu+au=f(|u|²)u, u∈H¹(R^N,C). (2.11) In turn solutions of (2.11) correspond to critical points of the functionalL^c_a:H¹(R^N,C)→R, defined by

L^c_a(v) = 1 2

R^N |∇v|²+a|v|² dy−

R^NF(|v|²)dy. (2.12)

In [43] the Pohozaev’s identity (2.9) and thus (2.10) is given for complex-valued solutions of (2.11). The following result relates the least energy levels of (2.7) and (2.11) and positively answers to a question of Kurata [35] (see also [42] for some elements of proof in that direction). WhenN = 2 we say that(f2) holds if

for allα >0 there existsC_α>0 such that|f(t²)| ≤C_αe^αt2, for allt≥0.

Lemma 2.3. Suppose that (f0)–(f2)and (f3) with mˆ =a hold and that N ≥2. Let E_a and E_a^c denote the least energy levels corresponding to equations (2.7)and(2.11). Then

E_a =E^c_a. (2.13)

Moreover any least energy solution of(2.11)has the forme^iτU whereU is a positive least energy solution of(2.7) andτ ∈R.

Proof. The inequality E_a^c ≤ E_a is obvious and thus to establish that E_a^c = E_a we just need to prove that E_a ≤E_a^c.

We know from [43] that each solution of (2.11) satisfies the Pohozaev’s identityP(u) = 0 whereP:H¹(R^N,C)→ Ris defined by

P(u) =N−2 2

R^N|∇u|²dy+N

R^Na|u|²

2 −F(|u|²)dy.

By Lemma 3.1 of [32] we have that

u∈Hinf¹(R^N,R) P(u)=0

L_a(u) =E_a. (2.14)

Also it is well known (see for example [31]) that for anyu∈H¹(R,C) one has

R^N

∇|u|²dy≤

R^N|∇u|²dy. (2.15)

Now let U be a solution of (2.11). If N = 2 we see from the definition of P that P(|U|) = 0 and from (2.15) thatL_a(|U|)≤L^c_a(U). ThusE_a≤E_a^c follows from (2.14). In addition, ifU is a least energy solution of (2.11),

necessarily

R^N

∇|U|²dy=

R^N|∇U|²dy (2.16)

and|U|is a least energy solution of (2.7). IfN ≥3 we see from (2.15) that either (i) P(|U|) = 0 andL_a(|U|) =L^c_a(U).

(ii) P(|U|)<0 and there existsθ ∈]0,1[ such that, for U_θ(·) =U(·/θ) we have P(|Uθ|) = 0. Then, since P(|Uθ|) = 0, it follows that

L_a(|Uθ|) = 1 N

R^N

∇|Uθ|²dy= θ^N−2 N

R^N

∇|U|²dy and thus

L_a(|Uθ|)< 1 N

R^N

∇|U|²dy≤ 1 N

R^N|∇U|²dy=L^c_a(U).

In both cases we deduce from (2.14) that E_a ≤E_a^c. In addition ifU is a least energy solution of (2.11) then (2.16) holds and in particular|U|is a least energy solution of (2.7).

Now, for any N ≥2, let U be a least energy solution of (2.11). Since |U| is a solution of (2.7) we get by elliptic regularity theory and the maximum principle that |U| ∈C¹(R^N,R) and |U|>0. At this point, using (2.16), the rest of the proof of the lemma is exactly the same as the proof of Theorem 4.1 in [31].

Remark 2.4. WhenN = 1 conditions which assure that (2.7) has, up to translation, a unique positive solution are given in [6] (see also [33] for alternative conditions). Now following the proof of Theorem 8.1.6 in [14] we deduce that any solution of (2.11) is of the form e^iθρwhereθ∈Randρ >0 is a solution of (2.7). Thus, under the assumptions of [6,33], the result of Lemma2.3also holds whenN= 1 and the positive least energy solution is unique.

Now letS_a be the set of least energy solutionsU of (2.11) satisfying

|U(0)|= max

y∈R^N|U(y)|.

By standard regularity any solution of (2.11) is at least C¹. Since f is not assumed to be locally H¨older continuous we do not know, in contrast to [5], if any least energy solution is radially symmetric. However the following compactness result can still be proved.

Proposition 2.5. For eacha >0 andN ≥3, S_a is compact in H¹(R^N,C). Moreover, there exist C,c > 0, independent of U ∈S_a, such that

|U(y)| ≤Cexp(−c|y|).

Proof. In [7], the same results are proved whenS_a is restricted to real solutions. Since, by Lemma2.3, any least energy solution of (2.11) is of the form e^iτU˜ with ˜U a least energy solution of (2.7) it proves the lemma.

3. Proof of Theorem 1.1

Let

δ= 1 10min

dist(M,R^N\O),min

i=j dist(O_i, O_j),dist(O,Z)

.

We fix a β ∈ (0, δ) and a cutoff ϕ ∈C₀^∞(R^N) such that 0 ≤ϕ ≤ 1, ϕ(y) = 1 for|y| ≤ β and ϕ(y) = 0 for

|y| ≥2β. Also, settingϕ_ε(y) =ϕ(εy) for eachx_i ∈(Mⁱ)^β andU_i∈S_mi,we define

U_ε^x1,...,xk(y) = k

i=1

e^{iA(xi)(y−xiε)}ϕ_ε

y−x_i ε

U_i

y−x_i ε

·

We will find a solution, for sufficiently smallε >0, near the set

X_ε={U_ε^x1,...,xk(y)|x_i∈(Mⁱ)^β andU_i∈S_mi for eachi= 1, . . . , k}. For eachi∈ {1, . . . , k} we fix an arbitraryx_i∈ Mⁱ and an arbitraryU_i∈S_mi and we define

W_εⁱ(y) = e^{iA(xi)(y−xiε)}ϕ_ε

y−x_i ε

U_i

y−x_i

ε ·

Setting

W_ε,tⁱ (y) = e^{iA(xi)(y−xiε)}ϕ_ε

y−x_i ε

U_i y

t −x_i εt

,

we see that lim_t→0W_ε,tⁱ ε = 0 (recall that N ≥ 3) and that Γ_ε(W_ε,tⁱ ) = Fε(W_ε,tⁱ ) for t ≥ 0. In the next proposition we shall prove that there existsT_i >0 such that Γ_ε(W_ε,Tⁱ _i)<−2 for anyε >0 sufficiently small.

Assuming this holds true, letγ_εⁱ(s) =W_ε,sⁱ fors >0 andγ_εⁱ(0) = 0.Fors= (s₁, . . . , s_k)∈T = [0, T₁]×. . .×[0, Tk] we define

γ_ε(s) = k i=1

W_ε,sⁱ _i and D_ε= max

s∈T Γ_ε(γ_ε(s)).

Finally for eachi∈ {1, . . . , k},letE_mi=L^c_mi(U) forU ∈S_mi. In what follows, we setE_m= min

i∈{1,...,k}E_mi and E=_k

i=1E_mi. For a setA⊂H_ε andα >0, we let A^α={u∈H_ε| u−Aε≤α}.

Proposition 3.1. We have (i) lim

ε→0D_ε=E;

(ii) lim sup

ε→0 max

s∈∂TΓ_ε(γ_ε(s))≤E˜= max{E−E_mi | i= 1, . . . , k}< E;

(iii) for each d >0,there existsα >0 such that for sufficiently smallε >0, Γ_ε(γ_ε(s))≥D_ε−αimplies that γ_ε(s)∈X_ε^d/2.

Proof. Since supp(γ_ε(s))⊂ M^2β_ε for eachs∈T,it follows that Γ_ε(γ_ε(s)) =Fε(γ_ε(s)) =_k

i=1Fε(γ_εⁱ(s)). Now, for eachi∈ {1, . . . , k}, we claim that

ε→0lim

R^N

∇

i −A_ε(y)

W_ε,sⁱ _i

²dy=s^N−2_i

R^N|∇U_i|²dy. (3.1)

Indeed

R^N

∇

i −A_ε(y)

W_ε,sⁱ _i ²dy=

R^N

|∇W_ε,sⁱ _i|²+|Aε(y)|²|W_ε,sⁱ _i|²−2 Re 1

i∇W_ε,sⁱ _i·A_ε(y)W_ε,sⁱ _i

dy (3.2) with

R^N|∇W_ε,sⁱ _i|²dy =

R^N

iA(x_i)U_i y

s_i − x_i εs_i

ϕ_ε

y−x_i

ε

+ 1 s_i∇Ui

y s_i − x_i

εs_i

ϕ_ε

y−x_i ε

+ε∇τϕ(εy−x_i)U_i y

s_i − x_i εs_i ²dy

=

R^N|A(xi)|² U_i

y s_i

²|ϕε(y)|²dy +

R^N

1 s_i∇U_i

y s_i

ϕ_ε(y) +ε∇τϕ(εy)U_i y

s_i ²dy. (3.3)

Moreover we have

R^N|Aε(y)|²|W_ε,sⁱ _i|²dy=

R^N|Aε(y)|² U_i

y s_i − x_i

εs_i ²ϕ_ε

y−x_i

ε

²dy (3.4)

and

R^NRe 1

i∇W_ε,sⁱ _i·A_ε(y)W_ε,sⁱ _i

dy=

R^NA_ε(x_i)·A_ε(y) U_i

y s_i − x_i

εs_i ²ϕ_ε

y−x_i

ε

²dy. (3.5)

Since, asε→0,

R^N|Aε(y)|² U_i

y s_i − x_i

εs_i ²ϕ_ε

y−x_i

ε

²dy→

R^N|A(xi)|² U_i

y s_i

²dy,

and

R^N

A_ε(x_i)·A_ε(y) U_i

y s_i − x_i

εs_i ²ϕ_ε

y−x_i

ε

²dy →

R^N|A(x_i)|² U_i

y s_i

²dy,

taking into account (3.2)–(3.5) it follows that,

R^N

∇

i −A_ε(y)

W_ε,sⁱ _i

²dy→ 1 s²_i

R^N

∇U_i y s_i

2

dy=s^N−2_i

R^N|∇U_i|²dy (3.6)

and this proves (3.1). Similarly using the exponential decay ofU_i we have, asε→0,

R^NV_ε(y)|W_ε,sⁱ _i|²dy→

R^Nm_i U_i

y s_i

²dy=m_is^N_i

R^N|U_i|²dy (3.7)

R^NF(|W_ε,sⁱ _i|²)dy→

R^NF U_i

y s_i

²

dy=s^N_i

R^NF(|U_i|²)dy. (3.8) Thus, from (3.1), (3.7) and (3.8),

Fε(γⁱ_ε(s_i)) = 1 2

R^N

∇

i −A_ε(y)

γ_εⁱ(s_i)

²dy+V_ε(y)|γ_εⁱ(s_i)|²dy−

R^NF(|γ_εⁱ(s_i)|²)dy

= s^N−2_i 2

R^N|∇U_i|²dy+s^N_i

R^N

1

2m_i|U_i|²−F(|U_i|²)dy+o(1).

Then, from the Pohozaev identity (2.9), we see that Fε(γ_εⁱ(s_i)) =

s^N−2_i

2 −N−2 2N s^N_i

R^N|∇U_i|²dy+o(1).

Also

t∈(0,∞)max t^N−2

2 −N−2 2N t^N

R^N|∇Ui|²dy=E_mi.

At this point we deduce that (i) and (ii) hold. Clearly also the existence of a T_i >0 such that Γ_ε(W_ε,Tⁱ _i)<−2 is justified. To conclude we just observe that forg(t) = ^tN−2₂ −^N_2N⁻²t^N,

g(t)

⎧⎪

⎨

⎪⎩

>0 fort∈(0,1),

= 0 fort= 1,

<0 fort >1,

andg(1) = 2−N <0.

Now let

Φⁱ_ε={γ∈C([0, T_i], H_ε)|γ(s_i) =γ_εⁱ(s_i) fors_i= 0 orT_i} (3.9) and

C_εⁱ= inf

γ∈Φⁱ_ε max

si∈[0,Ti]Γⁱ_ε(γ(s_i)).

For future reference we need the following estimate.

Proposition 3.2. Fori= 1, . . . , k,

lim inf

ε→0 C_εⁱ ≥E_mi.

Proof. Arguing by contradiction, we assume that lim inf_ε→0C_εⁱ < E_mi. Then, there existsα >0,ε_n →0 and γ_n∈Φⁱ_εn satisfying Γⁱ_εn(γ_n(s))< E_mi−αfors∈(0, T_i).

We fix anε_n>0 such that

m_i

2 ε^μ_n(1 + (1 +E_mi)^2/(p+2))<min{α,1} andFεn(γ_n(T_i))<−2 and denoteε_n byεand γ_n byγ.

SinceFε(γ(0)) = 0 we can finds₀∈(0,1) such thatFε(γ(s))≥ −1 fors∈[0, s₀] andFε(γ(s₀)) =−1. Then for anys∈[0, s₀] we have

Qⁱ_ε(γ(s))≤Γⁱ_ε(γ(s)) + 1≤E_mi−α+ 1

so that

R^N\Oⁱ_ε|γ(s)|²dy≤ε^6/μ 1 + (1 +E_mi)^2/(p+2)

∀s∈[0, s₀].

Now we notice that for anys∈[0, T_i],|γ(s)| ∈H¹(R^N,R) and by the diamagnetic inequality (2.3)

R^N

∇|γ(s)|²dy≤

R^N|D^εγ(s)|²dy. (3.10)

Then by (3.10) we have that fors∈[0, s₀] Fε(γ(s)) = 1

2

R^N|D^εγ(s)|²dy+m_i 2

R^N|γ(s)|²dy−

R^NF(|γ(s)|²) dy +1

2

R^N(V_ε(y)−m_i)|γ(s)|²dy

≥ 1 2

R^N

∇|γ(s)|²dy+m_i 2

R^N|γ(s)|²dy−

R^NF(|γ(s)|²) dy +1

2

R^N\Oⁱ_ε(V_ε(y)−m_i)|γ(s)|²dy

≥ 1 2

R^N

∇|γ(s)|²dy+m_i 2

R^N|γ(s)|²dy−

R^NF(|γ(s)|²) dy

−m_i 2

R^N\Oⁱ_ε|γ(s)|²dy

≥ 1 2

R^N

∇|γ(s)|²dy+m_i 2

R^N|γ(s)|²dy−

R^NF(|γ(s)|²) dy

−m_i

2 ε^6/μ 1 + (1 +E_mi)^2/(p+2)

= L_mi(|γ(s)|)−m_i

2 ε^6/μ 1 + (1 +E_mi)^2/(p+2)

. (3.11)

Thus, L_mi(|γ(s0)|)<0 and recalling that for the limiting equation (2.7) the mountain pass level corresponds to the least energy level (see [32]) we have that

s∈[0,Tmaxi]L_mi(|γ(s)|)≥E_mi. Then we infer that

E_mi−α ≥ max

s∈[0,Ti]Γⁱ_ε(γ(s))≥ max

s∈[0,Ti]Fε(γ(s))

≥ max

s∈[0,s0]Fε(γ(s))

≥ max

s∈[0,s0]L_mi(|γ(s)|)−m_i 2 ε^6/μ

1 + (1 +E_mi)^2/(p+2)

≥ E_mi−m_i 2 ε^6/μ

1 + (1 +E_mi)^2/(p+2)

(3.12)

and this contradiction completes the proof.