Multi-peak bound states for Schrödinger equations with compactly supported or unbounded potentials

www.elsevier.com/locate/anihpc

Multi-peak bound states for Schrödinger equations with compactly supported or unbounded potentials

Na Ba, Yinbin Deng, Shuangjie Peng

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, PR China Received 29 October 2009; received in revised form 28 April 2010; accepted 6 May 2010

Available online 25 May 2010

Abstract

In this paper, we will study the existence and qualitative property of standing wavesψ(x, t)=e^−iEtε u(x)for the nonlinear Schrödinger equationiε^∂ψ_∂t +_2m^ε2_xψ−(V (x)+E)ψ+K(x)|ψ|^p−1ψ=0,(t, x)∈R₊×R^N. LetG(x)= [V (x)]^{p+1p−1−N2} × [K(x)]^−p2−1and suppose thatG(x)hasklocal minimum points. Then, for anyl∈ {1, . . . , k}, we prove the existence of the standing waves inH¹(R^N)having exactlyllocal maximum points which concentrate nearllocal minimum points ofG(x)respectively as ε→0. The potentialsV (x)andK(x)are allowed to be either compactly supported or unbounded at infinity. Therefore, we give a positive answer to a problem proposed by Ambrosetti and Malchiodi (2007) [2].

©2010 Elsevier Masson SAS. All rights reserved.

MSC:35J20; 35J60

Keywords:Bound state; Multi-peak solutions; Nonlinear Schrödinger equation; Potentials compactly supported or unbounded at infinity

1. Introduction

In this paper, we will consider the existence of concentrating solutions for the following nonlinear Schrödinger equations

−ε²u+V (x)u=K(x)u^p, u >0, x∈R^N, u∈H¹

R^N

, (1.1)

whereε >0 is a small number, 1< p < (N+2)/(N−2)forN3, and 1< p <∞forN=1,2,V (x)0 has a compact support, and K(x)0 may tend to zero or infinity as |x| → ∞, H¹(R^N) is the usual Sobolev space {u∈D^1,2(R^N):

R^N|u|²<∞}.

A basic motivation for the study of (1.1) comes from looking for standing waves of the type ψ (x, t )=e^−iEtε u(x)

* Corresponding author.

E-mail address:sjpeng@mail.ccnu.edu.cn (S. Peng).

0294-1449/$ – see front matter ©2010 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2010.05.003

for the following nonlinear Schrödinger equations iε∂ψ

∂t = −ε²

2mxψ+

V (x)+E

ψ−K(x)|ψ|^p−2ψ, (t, x)∈R₊×R^N, (1.2) whereεis the Planck constant,iis the imaginary unit. Problem (1.2) arises in many applications. For example, in some problems arising in nonlinear optics, in plasma physics and in condensed matter physics, the presence of many particles leads one to consider nonlinear terms which simulate the interaction effect among them. The functionV (x) represents the potential acting on the particle andK(x)represents a particle-interaction term, which avoids spreading of the wave packets in the time-dependent version of the above equation. A solutionψis referred to as a bound state of (1.2) if ψ→0 as |x| → +∞. Furthermore, to describe the transition from quantum to classical mechanics, we letε→0 and thus the existence of solutionsψ_ε(x, t )of (1.2) for smallε(which is called semiclassic state) has an important physical interest.

There are a lot of works on problem (1.1). In [23], Floer and Weinstein considered the case N =1, p=3 and K(x)≡1. By using a Lyapunov–Schmidt reduction argument, they constructed a positive solutionu_εto problem (1.1) which concentrates around the critical point of potentialV (x) asε→0. Their method and results were later gen- eralized by Oh [31,32] to the higher-dimensional case with 1< p < (N+2)/(N −2)and multi-bump solutions concentrating near several non-degenerate critical points of V (x) as ε→0 were obtained. Existence of solutions concentrating at one or several points to problem (1.1) under different conditions has also been obtained in [4,6,12, 18–21,26,27,30,33,35–37].

We also mention some results on problem (1.1) in the case that lim inf_|x|→∞V (x) >0, Z =: {x ∈R^N: V (x)=0} = ∅, which was referred to as the critical frequency in [8]. In [8,9], it was shown that problem (1.1) admits a ground state concentrating on Z. Later, ifZ consists of several components, Cao and Peng in [15], Cao and Noussair in [13] and Cao, Noussair and Yan in [14] proved that problem (1.1) has multi-peak solutions which concentrate simultaneously on some prescribed components ofZor some points on whichV (x)does not vanish. For more results, we can refer to [10,11,34] and the references therein.

In all the works mentioned above, it is worth pointing out that the common assumptions are lim inf_|x|→∞V (x) >0 and 0< K(x) < c in R^N for some c >0. The assumption lim inf_|x|→∞V (x) >0 guarantees that the norm (

R^N|∇u|²+V (x)u²)^1/2is not weaker than the usual norm of Sobolev spaceH¹(R^N), and thus ensures that the solu- tions of (1.1) are bound states which are the most relevant from the physical point of view. Hence, by the boundedness ofK(x), the energy functionals corresponding to the equations are well defined inH¹(R^N)and the variational theory can be used directly. However, if the potential V (x)decays to zero or is compactly supported, orK(x) approaches infinity at infinity, variational theory inH¹(R^N)cannot be used, nor can one apply the perturbation methods, as in [4], since the spectrum of the linear operator−+V (x)is[0,+∞), see [7]. Therefore, the methods used in the previous papers cannot be employed any more in the present situation.

Recently, the case in whichV (x)andK(x)may decay to zero as|x| → ∞has been investigated. In [1], it is proved that if the potentialsV (x)andK(x)are smooth and satisfy

(V₀) ∃γ₀, γ₁>0: γ0

1+ |x|^α V (x)γ₁, (K₀) ∃β, k: 0< K(x) k

1+ |x|^β,

then (1.1) has a ground state solution for 0α <2 provided thatεis sufficiently small andσ < p < (N+2)/(N−2), where

σ= N+2

N−2−_α(N^4β₋₂₎, if 0< β < α,

1 otherwise.

In [3], this result was generalized to the case 1< p < (N+2)/(N−2)and 0α2. Moreover, setting G(x)=

V (x)^p+1

p−1−^N2

K(x)₋ ²

p−1,

which is referred to as a ground energy function introduced in [36], then it is verified that for any isolated stable sta- tionary pointx₀ofG(x), Eq. (1.1) has a bound state concentrating atx₀forεsufficiently small. WhenZ= {x∈R^N:

V (x)=0} = ∅andV (x),K(x)satisfy(V₀)and(K₀)at infinity respectively, Ambrosetti and Wang in [5] obtained a ground state of (1.1) which concentrates at some pointx^∗∈Z in the case 0α <2 andσ < p < ^N_N⁺₋²₂. Similar results can also be found in [10,11].

An open problem left, as proposed by Ambrosetti and Malchiodi in [2], is what happens if the potentialV (x) decays faster than |x|⁻² at infinity or has compact support? More precisely, it was illustrated in [2] the following:

“Is it possible to handle potentials with faster decay, or compactly supported? . . . . Clearly, the approach used so far cannot be repeated. However, any result, positive or negative, would be interesting.”

Concerning this open problem, some interesting results appeared recently. In [16], the caseV (x)∼ |x|^α at infinity withα−4 was considered by a constructive argument and multi-peak bound states with prescribed number of maximum points approaching to a local minimum point of the functionG(x)asε→0 were obtained. The case that V (x)has compact support, which is the most difficult case, was studied by Fei and Yin in [22], where it was shown that problem (1.1) admits a bound state with exact one local maximum pointx_εwhich tends to a minimum point of the functionG(x)asε→0. The method introduced in [22] is quite new. Precisely, the authors proved a mountain pass solution to a modified problem and then obtained a type of the weak Harnack inequality, by which they proved that the solution decays at infinity at the desired speed and hence is a bound state solving the original problem. However, the solution obtained in [22] is in some sense a least energy solution and indeed one-peaked. Moreover, because of the absence of an exact estimate to the energy related to the solution, their argument cannot be adopted to look for multi-peak solutions with higher energy. In this paper, we intend to focus on this problem.

We assume thatV (x), K(x)satisfy the following conditions:

(H₁).V (x)∈C(R^N)is compactly supported,V (x)0 andV (x)≡0;K(x)∈C(R^N),K(x)0.

(H2). There exist smooth bounded domainsΛ_i ofR^N, mutually disjoint such thatV (x) >0 andK(x) >0 onΛ¯_i, i=1, . . . , k, and

0< c_i≡ inf

x∈Λi

G(x) < inf

x∈∂Λi

G(x).

(H₃). There exist constantsk₁>0,β₁<2 such that 0K(x)k1

1+ |x|β1

inR^N. Our main result for Eq. (1.1) is as follows:

Theorem 1.1. Assume N 5 and max(1, (N+β1)/(N −2)) < p < (N +2)(N −2). Under the assumptions (H1)–(H3), there is anε0>0such that for every0< ε < ε0, problem(1.1)admits a positive solutionuε∈H¹(R^N).

Moreover,u_ε possesses exactlyk local maximax_εⁱ ∈Λ_i,i=1, . . . , k, which satisfy that G(x_εⁱ)→infx∈Λ_iG(x)as ε→0.

Remark 1.1. We actually prove the existence of solutions concentrating simultaneously at several points of a set {ξ₁, . . . , ξ_k}, where for eachi=1, . . . , k,ξ_i satisfiesG(ξ_i)=c_i.

Remark 1.2.From the proof of Theorem 1.1, we can see that Theorem 1.1 holds ifV (x)is a nonnegative continuous function inR^N. In particular, Theorem 1.1 remains true ifV (x)decays faster than|x|^−α including the case thatV (x) decays faster thane^−β|x|, whereα >0, β >0. Furthermore, ifV (x)=1/|x|²near infinity, which is related to the well-known Hardy inequality, our result is also true.

In order to obtain the existence of peaked solutions we use variational methods and penalization techniques. Since

R^NV (x)u²<∞cannot guarantee thatuis a bound state and

R^NK(x)|u|^p+1may not make sense foru∈H¹(R^N) due to the fact thatV (x) may vanish orK(x) may be unbounded at infinity, we will employ a penalization argu- ment which needs to modify the nonlinear termK(x)u^pin (1.1). To obtain multi-peak bound states with the desired number of peaks, we must give an exact estimate to the functional energy corresponding to the solutionu_ε of the modified problem. For this, we penalize the functional related to the modified problem by adding a penalizationP_ε(u) introduced in [19], which can also be used to prevent the concentration outside of ^k_i=1Λ_i. To ensure thatu_εsolves the original problem (1.1), we use an argument similar to [22] to obtain the weak Harnack inequality and thus find the decay rate ofu_ε outsideΛ_i. As we can see later, for the casek=1, our method can simplify the proof in [22].

Moreover, our result is true for the case thatV (x)decays faster than|x|^α(∀α <0)at infinity.

This paper is organized as follows: In Section 2 we define the modified functional needed for the proof of The- orem 1.1, and prove some preliminary results. Section 3 is devoted to the proof of Theorem 1.1. In this paper, the notationsC, C₁, . . .denote the generic positive constants depending only onV (x), K(x), p.

2. Preliminaries

LetΛ= ^k_i=1Λ_i. Forξ ∈Λ, consider the following equation −u(x)+V (ξ )u(x)=K(ξ )u^p(x), u(x) >0, x∈R^N,

u∈H¹ R^N

. (2.1)

The functional corresponding to (2.1) is defined as I_ξ(u)=1

2

R^N

|∇u|²+V (ξ ) 2

R^N

|u|²− K(ξ ) p+1

R^N

|u|^p+1. (2.2)

The following function G(ξ )= inf

u∈M^ξI_ξ(u) (2.3)

is referred to as ground energy function of (2.1) (see [36]), whereM^ξis the Nehari manifold defined by M^ξ=

u∈H¹ R^N

\ {0}:

I_ξ(u), u

=0 .

For further properties ofG(ξ ), one can see [36]. It is shown in [24,28] that, up to translations, (2.1) has a unique solutionU_ξ(x), which is radially symmetric and decays exponentially at infinity. Moreover,

G(ξ )=I_ξ(U_ξ). (2.4)

LetEεbe a class of weighted Sobolev space defined as follows

u∈D^1,2 R^N

R^N

ε²|∇u|²+V (x)|u|²

<∞

,

whereD^1,2(R^N)= {u∈L^N2N−2(R^N)| ∇u∈L²(R^N)}. The norm of the spaceEε is denoted by uε=

R^N

ε²|∇u|²+V (x)|u|^21/2 .

Under the assumptions(H₁)and(H₂), we can deduce from the Sobolev inequality the following:

Lemma 2.1.Assume that(H₁),(H₂)hold true, then for eachε∈(0,1], there exists a constantC₁>0independent ofεsuch that

Λ

K(x)|u|^p+1C1ε^{−N (p2−1)}u^pε⁺¹, ∀u∈E_ε. (2.5)

Next we define the first modification of our functional. Set f_ε(x, t )=min

K(x)

t⁺p

, ε³

1+ |x|^θ0t⁺, ε 1+ |x|^N

, wheret⁺=max{t,0}, andθ₀>2 will be suitably chosen later on.

Define

g_ε(x, ξ )=χ_Λ(x)K(x) ξ⁺p

+

1−χ_Λ(x)

f_ε(x, ξ ),

whereχ_Λ(x)represents the characteristic function of the setΛ. DenoteG_ε(x, u)=_u

0 g_ε(x, τ ) dτ.

Now we define the modified functionalL_ε:E_ε→Ras L_ε(u)=1

2

R^N

ε²|∇u|²+V (x)u²

− 1 p+1

Λ

K(x) u⁺p+1

−

R^N\Λ

F_ε(x, u),

whereF_ε(x, u)=(1−χ_Λ(x))_u

0 f_ε(x, τ ) dτ. Foru∈E_ε, due toθ₀>2, we see that

R^N\Λ

Fε(x, u)

R^N\Λ

ε³

1+ |x|^θ0u²Cε³

R^N\Λ

|u|^{N2N−2 N−2}

N

Cε³

R^N

|∇u|²Cεu²_ε. (2.6)

It follows from (2.5) and (2.6) thatL_ε(u)is well defined inE_ε.

Next, we introduce the second modification of the functional. AssumeG(ξ_i)=c_i,ξ_i∈Λ_i. Letσ_i>0 be such that sup

Λi

G(x)c_i+σ_i, (2.7)

and assume that k

i=1

σ_i<1

2min{c_i|i=1, . . . , k}. (2.8)

This can be achieved by makingΛ_ismaller if necessary. For mutually disjoint open setsΛ˜_i compactly containingΛ_i and satisfyingV (x) >0 on the closure ofΛ˜_i, we define onE_εthe functional

Lⁱ_ε(u)=1 2

Λ˜i

ε²|∇u|²+V (x)u²

− 1 p+1

Λi

K(x) u⁺p+1

−

Λ˜i\Λi

F_ε(x, u), (2.9)

and the penalization P_ε(u)=M

k

i=1

Lⁱ_ε(u)₊¹

2 −ε^N2(c_i+σ_i)¹²2

+. (2.10)

The constantMwill be chosen later. Now set J_ε(u)=L_ε(u)+P_ε(u).

Then it follows from Proposition A.1 in Appendix A that functionalJ_εis of classC¹. We show next thatJ_εsatisfies the Palais–Smale condition.

Lemma 2.2.Let{u_n}be a sequence inE_εsuch thatJ_ε(u_n)is bounded andJ_ε(u_n)→0. Then{u_n}has a convergent subsequence.

Proof. We first prove that the sequence{un}is bounded inEε. Similarly to (2.6), we have

R^N\Λ

fε(x, u)u

Cεu²ε, ∀u∈Eε. (2.11)

For a fixedq∈(2, p+1), by (2.6) and (2.11), we have

L_ε(u_n)−1 q

L_ε(u_n), u_n

= 1

2 −1 q

R^N

ε²|∇u_n|²+V (x)u²_n +

1 q − 1

p+1

Λ

K(x) u⁺_np+1

+1 q

R^N\Λ

f_ε(x, u_n)u_n−

R^N\Λ

F_ε(x, u_n)

C

R^N

ε²|∇u_n|²+V (x)u²_n

. (2.12)

Similarly, we find Lⁱ_ε(u_n)−1 q

Lⁱ_ε(u_n), u_n

C

˜ Λ_i

ε²|∇u_n|²+V (x)u²_n

. (2.13)

On the other hand, noting that P_ε(u_n), u_n

=M k

i=1

Lⁱ_ε(u_n)₊¹

2 −ε^N2(c_i+σ_i)¹²

+

Lⁱ_ε(u_n)₊₋¹

2

Lⁱ_ε(u_n), u_n , we derive from (2.13) that

P_ε(u_n)−1 q

P_ε(u_n), u_n

=M k

i=1

Lⁱ_ε(u_n)₊¹

2 −ε^N2(c_i+σ_i)¹²2 +

−1 qM

k

i=1

Lⁱ_ε(un)₊¹₂

−ε^N2(ci+σi)¹²

+

Lⁱ_ε(un)₊₋¹₂

Lⁱ_ε(un), un

=M k

i=1

Lⁱ_ε(u_n)₊¹

2 −ε^N2(c_i+σ_i)¹²

+

×

Lⁱ_ε(u_n)₊¹

2 −ε^N2(c_i+σ_i)¹²

+−1 q

Lⁱ_ε(u_n)₊₋¹

2

Lⁱ_ε(u_n), u_n

M

k

i=1

Lⁱ_ε(u_n)₊¹

2 −ε^N2(c_i+σ_i)¹²

+

×

Lⁱ_ε(u_n)₊¹

2 −ε^N2(c_i+σ_i)¹²

+−

Lⁱ_ε(u_n)₊¹

2

−M

k

i=1

ε^N2(c_i+σ_i)¹²

Lⁱ_ε(u_n)₊¹

2 −ε^N2(c_i+σ_i)¹²

+

−M¹² k

i=1

ε^N2(c_i+σ_i)¹²P

1

ε2(u_n)= −Cε^N2P

1

ε2(u_n).

Hence, by the fact thatP_ε(u_n)M

R^N(ε²|∇u_n|²+V (x)u²_n), we get C_εJ_ε(u_n)−1

q

J_ε(u_n), u_n

C

R^N

ε²|∇un|²+V (x)u²_n

−Cε^N2

R^N

ε²|∇un|²+V (x)u²_n¹²

, (2.14)

which implies that{u_n}is bounded inE_ε.

SinceE_ε→D^1,2(R^N) →H_loc¹ (R^N), the boundedness of{u_n}inE_ε implies that there existsu₀∈E_ε satisfying, after passing to a subsequence if necessary,

u_n→u₀ weakly inE_ε, u_n→u₀ strongly inL^q_loc R^N

, for 2q <2N/(N−2).

Now, to complete the proof, it suffices to prove thatunε→ u0ε asn→ ∞.

ByJ_ε(un), u0 →0, we see

R^N

ε²|∇u0|²+V (x)unu0

−

Λ

K(x) u⁺_np

u0−

R^N\Λ

fε(x, un)u0

+M k

i=1

Lⁱ_ε(u_n)

1

+2 −ε^N2(c_i+σ_i)¹²

+Lⁱ_ε(u_n)⁻

1

+2

×

˜ Λ_i

ε²|∇u0|²+V (x)unu0

−

Λ_i

K(x) u⁺_np

u0−

˜ Λ_i\Λ_i

fε(x, un)u0

=on(1). (2.15)

In addition, fromJ_ε(u_n), u_n =o_n(1)u_nand the boundedness of{u_n}, we have

R^N

ε²|∇u_n|²+V (x)u²_n

−

Λ

K(x) u⁺_np+1

−

R^N\Λ

f_ε(x, u_n)u_n

+M k

i=1

Lⁱ_ε(un)

1

+2 −ε^N2(ci+σi)¹²

+Lⁱ_ε(un)⁻

1

+2

×

˜ Λi

ε²|∇u_n|²+V (x)u²_n

−

Λi

K(x) u⁺_np+1

−

˜ Λi\Λi

f_ε(x, u_n)u_n

=o_n(1). (2.16)

On the other hand, we find

nlim→∞

R^N

V (x)u²_n= lim

n→∞

R^N

V (x)u_nu₀, (2.17)

nlim→∞

Λ

K(x) u⁺_np+1

= lim

n→∞

Λ

K(x) u⁺_np

u₀, (2.18)

nlim→∞

˜ Λ_i

V (x)u²_n= lim

n→∞

˜ Λ_i

V (x)u_nu₀, (2.19)

nlim→∞

Λ_i

K(x) u⁺_np+1

= lim

n→∞

Λ_i

K(x) u⁺_np

u0, (2.20)

nlim→∞

Λ˜i\Λi

f_ε(x, u_n)u_n= lim

n→∞

Λ˜i\Λi

f_ε(x, u_n)u₀, (2.21)

and for any fixed largeR >0 such thatΛ⊂BR(0),

nlim→∞

B_R(0)\Λ

fε(x, un)un= lim

n→∞

B_R(0)\Λ

fε(x, un)u0.

We will prove that for any givenδ >0, there existsR >0 such that for alln

R^N\B_R(0)

f_ε(x, u_n)u₀

< δ,

R^N\B_R(0)

f_ε(x, u_n)u_n < δ.

We just check the first inequality since the second one can be checked in a similar way. As in the proof of (2.6), we have

R^N\BR(0)

f_ε(x, u_n)u₀ ε³

R^N\BR(0)

|u_n||u₀| 1+ |x|^θ0 ε³

R^N\BR(0)

1 (1+ |x|^θ0)^N2

_N² u_n

L

2N

N−2(R^N)u₀

L

2N N−2(R^N)

ε

R^θ0−2u_nεu₀ε→0 asR→ ∞. So

nlim→∞

R^N\Λ

f_ε(x, u_n)u_n= lim

n→∞

R^N\Λ

f_ε(x, u_n)u₀. (2.22)

From the boundedness of{u_n}, we have M

Lⁱ_ε(u_n)

1

+2 −ε^N2(c_i+σ_i)¹²

+Lⁱ_ε(u_n)⁻

1

+2 =a_i+o_n(1), (2.23)

wherea_i0,i=1, . . . , kare constants. So inserting (2.17)–(2.23) into (2.15) and (2.16), we obtain o_n(1)=

R^N

ε²

|∇u_n|²− |∇u₀|² +

k

i=1

a_i+o_n(1)

˜ Λi

ε²

|∇u_n|²− |∇u₀|²

o_n(1).

Thus, we have limn→∞

R^N|∇u_n|²=

R^N|∇u₀|²and hence

nlim→∞

R^N

ε²|∇u_n|²+V (x)u²_n

=

R^N

ε²|∇u₀|²+V (x)u²₀ .

So, the proof of Lemma 2.2 is completed. 2

The previous lemma makes it possible to use the critical point theory to find critical points of the functionalJ_ε. We will formulate an appropriate minimax problem forJ_ε.

As in [17], we define a class of functionsΓ. A continuous functionγ: [0,1]^k→E_εis inΓ if there are continuous functionsg_i: [0,1] →E_εfori=1, . . . , ksatisfying

(i) supp{g_i(τ )} ⊂Λ_i for allτ∈ [0,1];

(ii) γ (τ₁, . . . , τ_k)=_k

i=1g_i(τ_i)for all(τ₁, . . . , τ_k)∈∂[0,1]^k; (iii) g_i(0)=0 andL_ε(g_i(1)) <0;

(iv) J_ε(γ (t ))ε^N(_k

i=1c_i−σ )for allt∈∂[0,1]^k, where 0< σ < ¹₂min{c_i|i=1, . . . , k}is a fixed number.

Define C_ε= inf

γ∈Γ sup

t∈[0,1]^k

J_ε γ (t )

.

As in [19], we can prove thatΓ is non-empty and

Lemma 2.3.

Cε=ε^N _k

i=1

ci+o(1)

.

Proof. The proof is similar to Lemma 2.3 in [15] (see also [19, Lemma 1.2]), and thus we omit it. 2

From Lemma 2.2 and Lemma 2.3, we conclude that there exists a critical pointu_ε∈E_εofJ_εsuch thatJ_ε(u_ε)=C_ε. We define the local weights

ρ_εⁱ=M

Lⁱ_ε(uε)₊¹₂

−ε^N2(ci+σi)¹²

+

Lⁱ_ε(uε)₊₋¹₂ , and then the function

ρ_ε= k

i=1

ρⁱ_εχ_Λ˜

i.

The critical pointu_εsatisfies in the weak sense ε²div

(1+ρε)∇uε

−(1+ρε)V (x)uε+(1+ρε)gε(x, uε)=0, (2.24)

and

R^N

(1+ρ_ε)

ε²∇u_ε∇ϕ+V (x)u_εϕ

=

R^N

(1+ρ_ε)g_ε(x, u_ε)ϕ, ∀ϕ∈E_ε. (2.25)

SetΛ^ε_i = {y∈R^N|εy∈Λi}andΛ˜^ε_i = {y∈R^N|εy∈ ˜Λi}. Letvε(y)=uε(εy)fory∈R^N, then we see div

1+ρε(εy)

∇vε

−

1+ρε(εy)

V (εy)vε+

1+ρε(εy)

gε(εy, vε)=0. (2.26)

Finally, proceeding as in the first part of the proof of Lemma 2.2, we obtain from the estimates onC_ε given in Lemma 2.3 that

R^N

ε²|∇u_ε|²+V (x)u²_ε

Cε^N,

R^N

|∇v_ε|²+V (εy)v_ε²

C. (2.27)

3. Proof of Theorem 1.1

In this section, we will prove thatρε≡0 anduεis indeed a solution of the original equation (1.1).

GivenR >0 and setA⊂R^N, we denote byNR(A)the set{y|dist(y, A) < R}. The following lemma means that vεis small away from the setΛ^ε= ^ki=1Λ^ε_i.

Lemma 3.1.There exists aC >0such that, for any givenR >0, one has

R^N\N_RΛ^ε

|∇v_ε|²+V (εy)v²_ε

C

R, (3.1)

forεsufficiently small.

Proof. For any givenR >0,ε >0, we may choose smooth cut-off functions 0ψ_i,R^ε 1 such that ψ_i,R^ε (y)=

1, if dist(y, Λ^ε_i) < R/2,

0, if dist(y, Λ^ε_i) > R, (3.2)

and|∇ψ_i,R^ε |4/R. Then setη_R^ε =1−k i=1ψ_i,R^ε .

Similarly to (2.6), we have

R^N\Λ^ε

η_R^ε2

f_ε(εy, v_ε)v_εε³

R^N\Λ^ε

(η^ε_Rv_ε)² 1+ |εy|^θ0 ε³

R^N\Λ^ε

η^ε_Rvε

_N^2N₋₂^N_N⁻²

R^N\Λ^ε

1 1+ |εy|^θ0

^N_2N²

ε³C

R^N\Λ^ε

∇ η_R^εv_ε²

R^N\Λ^ε

1 (1+ |εy|^θ0)^N2

²

N

=Cε

R^N\Λ^ε

∇ η^ε_Rv_ε²

R^N\Λ

1 (1+ |x|^θ0)^N2

²

N

=Cε

R^N\Λ^ε

∇ η^ε_Rv_ε²

Cε

R^N\Λ^ε

η^ε_R2

|∇v_ε|²+Cε

NR(Λ^ε)\Λ^ε

∇η^ε_R²v_ε².

Using the test function(η_R^ε)²v_εin (2.26), one gets C

R^N\N_R(Λ^ε)

|∇v_ε|²+V (εy)v²_ε

R^N\N_R(Λ^ε)

|∇v_ε|²+V (εy)v_ε²

−Cε

R^N\Λ^ε

η^ε_R2

|∇v_ε|²

−Cε

N_R(Λ^ε)\Λ^ε

∇η^ε_R²v²_ε

R^N\Λ^ε

1+ρ_ε(εy) η_R^ε2

|∇v_ε|²+V (εy)v²_ε−f_ε(εy, v_ε)v_ε

= −2

N_R(Λ^ε)\Λ^ε

1+ρε(εy)

vεη^ε_R∇vε∇η_R^ε. (3.3)

Note thatρ_ε is uniformly bounded by a constant possibly depending onM. Using this, the choice ofη^ε_R,(H₁),(H₂) and (2.27), we find that (3.1) follows immediately from (3.3). 2

Before we proceed further, we give a preliminary lemma.

Lemma 3.2.Assume thatv∈H¹(R^N)∩C(R^N)is nonnegative and satisfies the equation v−v+χ_{x1<0}v^p=0, x∈R^N.

Thenv≡0.

Proof. Standard regularity arguments yieldv∈C¹(R^N)and v→0,∇v→0 as|x| → +∞. Using _∂x^∂v₁ as a test function, we see

1 2

R^N−1

dx +∞

−∞

∂

∂x₁

|∇v|²+v²

dx₁− 1 p+1

R^N−1

v^p+1 0, x

dx=0.