Standing Waves for Nonlinear Schrödinger Equations With Singular Potentials

www.elsevier.com/locate/anihpc

Standing waves for nonlinear Schrödinger equations with singular potentials

Ondes stationnaires pour les équations nonlinéaires de Schrödinger avec potentiels singulaires

Jaeyoung Byeon

, Zhi-Qiang Wang

aDepartment of Mathematics, POSTECH, San 31 Hyoja-dong, Nam-gu, Pohang, Kyungbuk 790-784, Republic of Korea bDepartment of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA

Received 14 November 2007; accepted 28 March 2008 Available online 7 May 2008

Abstract

We study semiclassical states of nonlinear Schrödinger equations with anisotropic type potentials which may exhibit a combi- nation of vanishing and singularity while allowing decays and unboundedness at infinity. We give existence of spike type standing waves concentrating at the singularities of the potentials.

©2008 Elsevier Masson SAS. All rights reserved.

Résumé

Nous étudions les états semi-classiques des équations de Schrödinger non linéaires avec potentiels de type anisotropiques qui peuvent tendre vers zéro à l’infini, pour lesquels des phénomènes d’évanescence et de singularité sont possibles. Nous donnons l’existence d’ondes stationnaires se concentrant aux singularités des potentiels.

©2008 Elsevier Masson SAS. All rights reserved.

Keywords:Nonlinear Schrödinger equations; Singularities of potentials; Decaying and unbounded potentials

1. Introduction

This paper is concerned with standing waves for nonlinear Schrödinger equations ih¯∂ψ

∂t + ¯h²

2 ψ−V (x)ψ+K(x)|ψ|^p−1ψ=0,

* Corresponding author.

E-mail address:zhi-qiang.wang@usu.edu (Z.-Q. Wang).

1 This work of the first author was supported by the Korea Research Foundation Grant (KRF-2007-313-C00047).

0294-1449/$ – see front matter ©2008 Elsevier Masson SAS. All rights reserved.

doi:10.1016/j.anihpc.2008.03.009

whereh¯ denotes the Planck constant,iis the imaginary unit. The equation arises in many fields of physics, in particu- lar, when we describe Bose–Einstein condensates (refer [18,19]) and the propagation of light in some nonlinear optical materials (refer [20]). In this paper we are concerned with the existence of standing waves of the nonlinear Schrödinger equation for smallh.¯ For smallh >¯ 0,these standing wave solutions are refereed as semiclassical states. Here a so- lution of the formψ (x, t )=exp(−iEt /h)v(x)¯ is called a standing wave. Then, a functionψ (x, t )≡exp(−iEt /h)¯ v(x)is a standing wave solution if and only if the functionvsatisfies

¯ h²

2 v−

V (x)−E

v+K(x)|v|^p−1v=0, x∈Rⁿ.

With a simple re-scaling and renaming the potentialV −Eto beV we work on the following version of the equation in this paper

−ε²v+V (x)v=K(x)v^p, v >0, x∈Rⁿ,

v∈W^1,2(Rⁿ), lim_|x|→∞v(x)=0. (1)

Hereε is a small parameter,V , K are nonnegative potentials, andpis subcritical 1< p < ⁿ_n⁺₋²₂ with 2^∗=_n²ⁿ₋₂ the critical exponent for n3. In recent years intensive works have been done in understanding solutions structure of Eq. (1) as ε→0.One of the most characteristic feature is that the semiclassical bound states exhibit concentration behaviors asε→0 (see the classical work [15] by Floer and Weinstein, [21,13,17,14,3,7] and the recent monograph [2] and references therein). In particular, some recent works have been devoted to the cases where the potentials may have vanishing points and may be decaying to zero at infinity [1,4–6,8–11,22]. In [1] ground state solutions in the associated weighted Sobolev spaces are obtained for positive potentials with decay at infinity. In [4,5] decaying potentials are also considered and spike solutions which concentrate at points of positive potential values are given.

Then in [6] ground state solutions concentrating near zeroes of potentials were constructed in the weighted Sobolev spaces.

In this paper we consider concentration solutions which both concentrate near zeroes of the potentialV and sin- gularities of the potentialsV andKfor Eq. (1). The potentialsV may decay at infinity andKmay be unbounded at infinity. One feature of our results is that the solutions we construct have small magnitudes comparing with the spike solutions concentrating at points of positive potential values. Another feature is that these solutions may have very different limiting equations under quite different scalings.

We assume thatV satisfies

(V) V ∈L¹_loc(Rⁿ)∩C(Rⁿ\S0,[0,∞)) for a bounded Lebesgue measure zero set S0, Z ∩S0= ∅ where Z= {x∈Rⁿ\S0|V (x)=0}; lim infdist(x,S0)→0V (x)∈(0,∞]; lim inf_|x|→∞|x|²V (x)4λ >0 for someλ >0.

We assume thatKsatisfies

(K) K∈L^q_loc⁰(Rⁿ)for someq₀>_2n−(p+²ⁿ_1)(n−2), n3;K∈C(Rⁿ\S,[0,∞))for a bounded and Lebesgue measure zero setS; lim sup_|x|→∞K(x)|x|^−γ∞<∞for someγ_∞>0.

If S is a singleton, for example S= {0}, the first condition of (K) holds if lim sup_|x|→0 |x|^γK(x) <∞ for some γ <^2n−(p+₂^1)(n−2).

Our main existence result is the following one.

Theorem 1.Letn3.Suppose that(V)and(K)hold. LetA⊂Z∪Sbe an isolated compact subset ofZ∪S such thatA∩S0∩S0\A= ∅and

0<dist(x,A)lim →0V^{2n−(p+21)(n−2)}(x)/K²(x)=0.

Then forεsufficiently small,(1)has a positive solutionwε∈W^1,2(Rⁿ)∩L^∞(Rⁿ)such that

lim→0wε∞=0 and lim inf

ε→0 ε^p−2+1wε∞>0. (2)

Moreover, for eachδ >0,there are constantsC, c >0such that w_ε(x)Cexp

−c ε

1+dist

x, A^δ₋^√λ/ε

, x /∈A^δ, (3)

whereA^δ≡ {x∈Rⁿ|dist(x, A)δ}.

We also study the asymptotic profile of the concentrating solutions given above. It turns out the asymptotic behavior depends on the local behavior of the potentialsV andKnear the concentrating setA. We give some detained results in Section 3.

Our scheme for a proof of the existence of localized solutions in Theorem 1 is as follow. Since there are singular points ofK, andV may converges to 0 at infinity, we consider a truncated equation on a bounded ballB(0, μ)with a truncatedK^μinstead ofKand a homogeneous Dirichlet boundary condition. For the truncated problem, we consider a minimization problem with two constraints, where we should delve an appropriate weight and an appropriate exponent for a constraint. The existence of a minimizeru^μ_ε follows from our well chosen setting. Then, we get an upper bound estimate and a lower bound estimate for the minimum. One crucial step in our argument is to obtain an exponential decay, uniform for largeμ, ofu^μ_ε on a certain set. The decay estimate is derived by combining Moser iterations, standard elliptic estimates, Caffarelli–Kohn–Nirenberg inequalities and comparison principles. Then, we show that a scaled minimizerw_ε^μis a solution of the truncated problem for smallε >0 and largeμ >0,and that a weak limitw_ε of{w^μ_ε}μis a desired solution of our original problem.

We close up the introduction with some discussions of known results and comparison with our new results. During the last twenty years there have been intensive work on semiclassical states of standing waves for the nonlinear Schrödinger equations with potentials. Spike solutions concentrating at points of positive potentials values forV and Khave been given in the pioneer work [15] and subsequent works e.g., [1,4,5,21,22]. These solutions are shown to have nice asymptotic behavior in the sense that ifw_ε(x)is a solution of (1) andx₀is the concentration point then W_ε(x)=w_ε(ε(x−x₀)) converges uniformly to a least energy solution of the following limiting equation U+ V (x₀)U =K(x₀)U^p,U >0,x∈Rⁿ. ThusU is the limiting profile of semiclassical limits in this case. In [9–11]

the authors have studied the critical frequency case for which at the concentration pointx₀the potential V is zero, i.e.,V (x₀)=0. We have found that for this case the limiting equations are abundant and appear in different families and that under different scalings the limiting profile of the semiclassical states have small magnitudes, i.e.,w_εL^∞

tends to zero asε→0. The new phenomenon we present in the current paper here is that the concentration for spike solutions can be at zeroes and singularities of the potentials for bothV andK. It depends on the zero set of a weighted potential involving bothV andK. Comparing with the results in [9,10] we construct small solutions even whenV (x₀) is positive. On the other hand our new results here allow us to construct solutions concentrating at singular points of V andK. We also investigate the limiting profile of these localized solutions. Under more precise information on the local behaviors ofV andK near the concentration points we derive a variety of limiting equations. One simple example covered by our results is whenV (x)= |x|^τ for|x|small,V (x)|x|⁻²for|x|large andK(x)= |x|^−γ for

|x|small,K(x)|x|^γ∞ for|x|large for someγ_∞>0. Then forτ >−2, γ∈(−τ (2n−(p+1)(n−2))/4, (2n− (p+1)(n−2))/2), our result applies to give the existence of a localized solutionw_εwhich under a suitable scaling converges to a least energy solution of Eq. (27) (see the statement of Theorem 11).

The proof of Theorem 1 is given in Sections 2 and 3 is devoted to asymptotic analysis of the localized solutions as ε→0.

2. Proof of Theorem 1

For a proof of our main results we further elaborate the minimization techniques in [9,11] to construct the spike solutions concentrating near zeroes and singularities of the potentials.

LetA⊂Z∪Sbe the isolated set assumed in the theorem. We chooseδ∈(0,1)such thatA^8δ∩((Z∪S)\A)= A^8δ∩(S0\A)= ∅, where ford >0, A^d= {x∈Rⁿ|dist(x, A)d}. We denote in the followingA^d_ε = {x∈Rⁿ|εx∈ A^d}forε, d >0.LetE_εbe the completion ofC₀^∞(Rⁿ)with respect to the norm

uε=

ε²|∇u|²+V (x)u² 1/2

.

We define a truncated function forμ >0 K_μ(x)=

min{K(x), μ} ifx∈Rⁿ\A^4δ, K(x) ifx∈A^4δ. We consider the following problem forμ >0

ε²v−V (x)v+K_μ(x)v^p=0, v >0, x∈B(0, μ),

v(x)=0 on∂B(0, μ). (4)

DefineE_ε^μ≡(C₀^∞(B(0, μ)), · ε)and chooseR₀>0 so thatZ∪S0∪S⊂B(0, R0/2). Note that n

2 < 2n

2n−(n−2)(p+1), n

2<p+1 p−1. From now, we fix a numberβ >0 satisfying

n

2 β <min

2n

2n−(n−2)(p+1),p+1 p−1

. (5)

For a sufficiently largeα >0 which will be specified later, we defineχ_εby

χ_ε^μ(x)=

⎧⎪

⎪⎨

⎪⎪

⎩

ε^−α if|x|R₀andx /∈(Z∪S)^4δ, ε^−α(max{1, Kμ(x)})^β ifx∈((Z∪S)\A)^4δ, ε^−2α|x|^α if|x|R0,

0 ifx∈A^4δ.

We defineβ˜≡max{β(p−1),2}.Defining Φ_ε^μ(u)≡

Rⁿ

K_μ|u|^p+1dx and Ψ_ε^μ(u)≡

Rⁿ

χ_ε^μ|u|^β˜dx, we consider the following minimization problem

M_ε^μ=inf

u²_εΦ_ε^μ(u)=1, Ψ_ε^μ(u)1, u∈E_ε^μ

. (6)

Since(K)holds andχε^μ=0 inA^4δ,we see thatΦε^μ, Ψε^μ∈C¹(Eε^μ).

Lemma 2.limε→0ε⁻ⁿ

p−1

p+1M_ε^μ=0uniformly for largeμ >0.

Proof. Note that M_ε^μ inf

u∈C^∞₀ (A^4δ)

ε²|∇u|²+V (x)u²dx (

K(x)|u|^p+1dx)^p+21 . Lettingv(x)=u(εx)we have

M_ε^με

n(p−1) p+1 inf

v∈C₀^∞(A^4δ_ε)

|∇v|²+V (εx)v²dx (

K(εx)|v|^p+1dx)^p2+1 .

For anyx₀∈A^4δ\A,we can taker >0 withB(x₀, r)⊂A^4δ\Asuch thatV (x)2V (x0)forx∈B(x₀, r),and K(x)¹₂K(x₀)forx∈B(x₀, r).Then, we see easily that

M_ε^με

n(p−1)

p+1 inf

v∈C₀^∞(B(x₀/ε,r/ε)

|∇v|²+V (εx)v²dx (

K(εx)|v|^p+1dx)^p+21

ε

n(p−1)

p+1 inf

v∈C^∞₀ (B(0,r/ε))

|∇v|²+2V (x0)v²dx ( ₁

2K(x0)|v|^p+1dx)^p+21

=ε

n(p−1) p+1

2V (x0)2n−(p+1)(n−2) 2(p+1)

K(x₀)/2₋ ²

p+1inf |∇v|²+v²dx (

|v|^p+1dx)^p+21

v∈C₀^∞ B

0,

2V (x0)r/ε .

Thus, it follows that there existsC >0 such that for anyx₀∈A^4δ\Asufficiently close toA,

εlim→0ε^−np−1p+1M_ε^μCV2n−(p+1)(n−2)

2(p+1) (x0)/K^p2+1(x0).

Then, the estimate follows. 2

Lemma 3.The minimizationM_ε^μis achieved by a nonnegativeu^μ_ε ∈E^μ_ε which satisfies for someη^μ_ε >0ξ_ε^μ

−ε²u^μ_ε +V (x)u^μ_ε =η_ε^μKμ(x) u^μ_εp

+ξ_ε^μχ_ε^μ(x) u^μ_εβ˜−1

inB(0, μ). (7)

Proof. From the choice of β and the definitionK^μ andχ_ε^μ,we see thatM_ε^μis achieved by a minimizeru^μ_ε ∈E_ε^μ which can be assumed to be nonnegative and satisfies Eq. (7) with Lagrange multipliersα_ε^μandβ_ε^μ. Following an argument from [9] we haveηε^μ>0ξε^μ. 2

Lemma 4.

εlim→0ε⁻ⁿ

p−1

p+1η_ε^μ=0 uniformly for largeμ >0.

Proof. By contradiction we assume there exist η₀>0, μ_m→ ∞, ε_m →0 such that lim infm→∞ε⁻ⁿ

p−1

m p+1η^μ_m^m η₀>0. Letu_m=u^μ_εm^m andη_m=η^μ_εm^m. For smallb >0,we define a cut-off functionφ_b(x)which is 1 forx satis- fying dist(x,Rⁿ\A^4δ) > band 0 forx /∈A^4δ, and|∇φ|2/b. Multiplying Eq. (7) byφbumand integrating over the space, and using the fact that inf_{x∈A4δ|d(x,Rⁿ\A^4δ)b}V (x)c0>0 (independent of smallb), we get that for some C >0,

ε⁻ⁿ

p−1 p+1

m η_m

{x|d(x,Rⁿ\A^4δ)>b}

K_μmu^p_m⁺¹Cε⁻ⁿ

p−1 p+1

n ε²_m|∇u_m|²+V (x)u²_m .

We see from Lemma 2 that the right-hand side in above inequality converges to 0 asn→ ∞.Then we see that

mlim→∞

{x|d(x,Rⁿ\A^4δ)>b}

K_μmu^p_m⁺¹=0. (8)

On the other hand, letψ be another cut-off function satisfying thatψ (x)=1 forx∈A^4δ\A^3δ andψ (x)=0 for x ∈A^2δ or x /∈A^5δ, and |∇ψ|2/δ. Note thatβ < p˜ +1. Then, using (8), the fact Ψ_ε^μ_m^m(u_m)1 and Hölder inequality, we see that

RⁿK_μm(ψ u_m)^p+1→1 asm→ ∞. Considerw_m(x)=ε

p+1n

m ψ (ε_mx)u_m(ε_mx)and by using Lemma 2 we havewm→0 inH¹(Rⁿ). By embedding theorems and the fact inf_x∈A5δ\A^2δV (x) >0,it follows that Kμ_m(εmx)w^p_m⁺¹→0. This contradicts that

Kμ_n(εmx)w_m^p+1=

Kμ_m(x)(ψ (x)um(x))^p+1→1 asm→ ∞. 2 Lemma 5.Ifα >0is sufficiently large,

lim inf

ε→0 ε⁻²η^μ_ε >0 uniformly for largeμ >0.

Proof. Arguing indirectly, we assume for a subsequence (still denoted byε)ε⁻²η^μ_ε →0. Choose a cut-off functionφ satisfyingφ(x)=1 forx∈A^4δandφ(x)=0 forx /∈A^5δ. Then for some constantC >0 independent of smallε >0 and largeμ >0,it follows that

∇

φu^μ_ε²dx2

φ²∇u^μ_ε²+ |∇φ|² u^μ_ε2

dx C ∇u^μ_ε²+ε⁻²V

u^μ_ε2

dx=Cε⁻²M_ε^μ Cε⁻²η^μ_ε →0 asε→0.

Then, by Hölder’s inequality and Sobolev embedding, we see that

A^4δ

K_μ u^μ_εp+1

=

Rⁿ

K

φu^μ_εp+1

→0 asε→0.

By the constraintΨ_ε^μ(u^μ_ε)1,we see that

{x /∈(Z∪S)^4δ||x|R0}

u^μ_εβ˜

dxε^α, (9)

{x∈(Z∪S\A)^4δ}

K_μ^β u^μ_εβ˜

dxε^α (10)

and

{|x|R0}

|x|^α u^μ_εβ˜

dxε^2α. (11)

Note thatβ < p˜ +1.Then, using Hölder’s inequality, we see that for someC >0,independent ofμ, ε >0,

{x∈(Z∪S\A)^4δ}

K_μ u^μ_εp+1

dx

{x∈(Z∪S\A)^4δ}

(K_μ)^β

u^μ_εβ(p−1)

dx

1/β

{x∈(Z∪S\A)^4δ}

u^μ_ε2β/(β−1)(β−1)/β

C

{x∈(Z∪S\A)^4δ}

(Kμ)^β

u^μ_εβ(p−1)

dx

1/β

{x∈(Z∪S\A)^4δ}

u^μ_ε2^∗2/2^∗

.

Thus, ifβ˜=β(p−1),it follows that

εlim→0

{x∈(Z∪S\A)^4δ}

K_μ u^μ_εp+1

dx=0.

Whenβ˜=2,we deduce from conditions (5) and (10) that for someC >0,

{x∈(Z∪S\A)^4δ}

(Kμ)^β

u^μ_εβ(p−1)

dx

{x∈(Z∪S\A)^4δ|u^με(x)ε^α/2}

(K_μ)^β

u^μ_εβ(p−1)

dx+

{x∈(Z∪S\A)^4δ|u^με(x)ε^α/2}

(K_μ)^β

u^μ_εβ(p−1)

dx

ε^{αβ(p−1)/2}

{x∈(Z∪S\A)^4δ}

(K_μ)^βdx+ε^{αβ(p−1)/2}

{x∈(Z∪S\A)^4δ|u(x)^μ_εε^α/2}

(K_μ)^β

u^μ_ε/ε^α/2β(p−1)

dx

ε^{αβ(p−1)/2}C+ε^{αβ(p−1)/2−α}

{x∈(Z∪S\A)^4δ}

(K_μ)^β u^μ_ε2

dx ε^{αβ(p−1)/2}C+ε^{αβ(p−1)/2}.

Thus, we see that

εlim→0

{x∈(Z∪S\A)^4δ}

K_μ u^μ_εp+1

dx=0

uniformly forμ >0.Using Hölder’s inequality, condition (K), (9) and (11), we see that ifα >0 is sufficiently large,

εlim→0

{x /∈(Z∪S)^4δ||x|R₀}

K_μ u^μ_εp+1

dx=0

and

εlim→0

{|x|R0}

K_μ u^μ_εp+1

dx=0 uniformly for largeμ >0.Then, we get

εlim→0

K_μ

u^μ_εp+1

dx=0, which contradicts the constraint

Kμ(u^με)^p+1dx=1.This completes the proof. 2 We denotew^μ_ε ≡(η^μ_ε)^p−11u^μ_ε.Then, we see that

−ε²w_ε^μ+V w_ε^μK_μ w^μ_εp

inB(0, μ). (12)

From Lemmas 2 and 4, we see that

εlim→0ε⁻ⁿ

Rⁿ

ε²∇w_ε^μ2+V w_ε^μ2

=0 uniformly for largeμ >0. (13)

Then, it follows from Sobolev embedding and the factΨε^μ(u^με)1 that for anyq∈ [ ˜β,2^∗)andr∈(0, δ),

εlim→0ε⁻ⁿ

Rⁿ\A^r

w_ε^μq

=0 uniformly for largeμ >0, (14)

where 2^∗=2n/(n−2)forn3.

Lemma 6.For anyr∈(0, δ),there existc, C >0,independent of largeμ >0,such that for smallε >0, w^μ_ε(x)Cexp(−c/ε) for|x|R₀ and dist(x,Z∪S) > r.

Proof. Note that in the setB(0, R0)\{x|dist(x,Z∪S) > r},V has a positive lower bound andKis continuous (thus has a upper bound). We may use (13) and (14) together with elliptic estimates (refer [16]) and a maximum principle argument similar to [9] (Lemma 2.6, 2.7 there) to deduce the estimate. 2

Lemma 7.There existc, C >0,independent of largeμ >0,such that for smallε >0, w^μ_ε(x)Cexp(−c/ε) forx∈

(Z∪S)\Aδ

. Proof. Denotingw^μ_ε ≡(η^μ_ε)^p−11u^μ_ε,we see that

−ε²w_ε^μ+V (x)w^μ_ε K_μ w_ε^μp

inB(0, μ). (15)

Letφ∈C^∞₀ ((Z∪S\A)^2δ)be a cut-off function such thatφ(x)=1 forx∈(Z∪S\A)^δand|∇φ|4/δ. Multiplying both sides of (15) through by(w_ε^μ)^2l+1φ²withl0, we see that

ε² l+1

Rⁿ

∇

w^μ_εl+1φ²dx ε² l+1

Rⁿ

w_ε^μ2l+2

|∇φ|²dx+

Rⁿ

K_μ

w_ε^μp−1 w_ε^μ2l+2

φ²dx.

Then, by the Sobolev inequality and Hölder’s inequality, it follows that for somec >0, independent ofφ, l, ε, μ

cε²

l+1w^μ_ε2l+2

φ²

L^n/(n−2) ε² l+1

Rⁿ

w^μ_ε2l+2

|∇φ|²dx

+

supp(φ)

(K_μ)^β

w^μ_εβ(p−1)

dx 1/β

w^μ_ε2l+2

φ²

L^β/(β−1). (16)

Ifβ(p−1)2,it follows that

supp(φ)

(K_μ)^β

w^μ_εβ(p−1)

dxε^α.

Whenβ(p−1) <2,it follows that for someC >0,independent of smallε >0 and largeμ >0,

supp(φ)

(K_μ)^β

w^μ_εβ(p−1)

dx

=

{x∈supp(φ)|w^μ_ε(x)ε^α/2}

(K_μ)^β

w^μ_εβ(p−1)

dx+

{x∈supp(φ)|w_ε^μ(x)>ε^α/2}

(K_μ)^β

w_ε^μβ(p−1)

dx

Cε^{αβ(p−1)/2}+ε^{αβ(p−1)/2}

{x∈supp(φ)|w^με(x)>ε^α/2}

(K_μ)^β

w_ε^μ/ε^α/2β(p−1)

dx

Cε^{αβ(p−1)/2}+ε^{αβ(p−1)/2}

supp(φ)

(Kμ)^β

w^μ_ε/ε^α/22

dx Cε^{αβ(p−1)/2}+ε^{αβ(p−1)/2}

η^μ_ε2/(p−1)

(C+1)ε^{αβ(p−1)/2},

where we used the factΨ_ε^μ(u^μ_ε)1.Then we deduce that there existsC, c >0,independent ofl, ε, μ,satisfying w_ε^μ2l+2

φ²

L^n/(n−2)Cexp(−c/ε)+C(l+1)ε

αmin{1,β(p−1)/2}

β −2w^μ_ε2l+2

φ²

L^β/(β−1). (17)

Note that _β^β₋₁<_n−ⁿ₂.Then, by Hölder inequality again there is a constantC₁only depending onn, βandδsuch that w_ε^μ2l+2

φ²

L^n/(n−2)Cexp(−c/ε)+C1(l+1)ε

αmin{1,β(p−1)/2}

β −2w_ε^μ2l+2

φ²

L^n/(n−2). (18)

We take largeα >0 so that^αmin{1,β(p_β ^−1)/2}>2.This implies that for any largeq >0,there existsC, c >0 such that for smallε,independent ofμ >0,

(Z∪S\A)^δ

w^μ_εq

dxCexp(−c/ε).

Applying an elliptic estimate [16] to (15), we see that for anys >2 andt > n/2,there exists a constantC >0, independent ofε, μ >0,satisfying

w_ε^μ

L^∞((Z∪S\A)^δ)w_ε^μ

L^s((Z∪S\A)^2δ)+

(Z∪S\A)^2δ

K^μt w_ε^μpt

dx 1/t

.

Note thatK^μKandK∈L^q_loc⁰ for someq0>_2n−(p+²ⁿ_1)(n−2) >ⁿ₂. We taket∈(n/2, q0).This implies thatK^t∈L^s_loc for somes >1. Thus, we see from Hölder inequality that for someC,independent of largeμand smallε >0,

w_ε^μ

L^∞((Z∪S\A)^δ)Cexp(−c/ε).

Thus, for someC, c >0,independent of largeμ >0 and smallε >0,we see that w_ε^μ(x)Cexp(−c/ε) forx∈

(Z∪S)\Aδ

. 2

The last two lemmas show the following estimate

Lemma 8.For anyr∈(0, δ),there existc, C >0,independent of largeμ >0,such that for smallε >0, w^μ_ε(x)Cexp(−c/ε) for|x|R0 and dist(x, A) > r.

Lemma 9.There existc, C >0,independent of largeμ >0,such that for smallε >0, w^μ_ε(x)Cexp(−c/ε)|x/R₀|^−√λ/ε forR₀|x|μ.

Proof. First we see from condition (K) that there is a constantC >0,independent of largeμ >0,satisfying

−ε²w_ε^μ(x)+V (x)w_ε^μ(x)C|x|^γ∞

w_ε^μ(x)p

onB(0, μ)\B(0, R0). (19)

For anyϕ∈C₀^∞(Rⁿ\B(0, R0),[0,1]), a >0 andb0,we multiply both sides of (19) through by|x|^a(w_ε^μ)^2b+1ϕ² and integrate by parts. Then, we deduce that

ε²

|x|^a∇ w^μ_εb+1

ϕ²dxε²

|x|^a

w^μ_ε2b+2

|∇ϕ|²+a|x|^a−1∇

w_ε^μb+1w^μ_εb+1

ϕ²dx +C1(b+1)

|x|^a+γ∞

w_ε^μp+2b+1

ϕ²dx ε²

|x|^a

w^μ_ε2b+2

|∇ϕ|²+aε²

|x|^a−1

w^μ_ε2b+2

|∇ϕ|ϕ dx +aε²

|x|^a/2∇

w^μ_εb+1ϕ|x|^a/2−1 w_ε^μb+1

ϕ dx +C1(b+1)

|x|^a+γ∞

w_ε^μp+2b+1

ϕ²dx ε²

|x|^a

w^μ_ε2b+2

|∇ϕ|²+aε²

|x|^a−1

w^μ_ε2b+2

|∇ϕ|ϕ dx +aε²

1 2a

|x|^a∇ w^μ_εb+1

ϕ²+a 2

|x|^a−2

w^μ_ε2b+2

ϕ²dx +C1(b+1)

|x|^a+γ∞

w_ε^μp+2b+1

ϕ²dx.

This implies that ε²

|x|^a∇ w^μ_εb+1

ϕ²dx2ε²

|x|^a

w^μ_ε2b+2

|∇ϕ|²+2aε²

|x|^a−1

w^μ_ε2b+2

|∇ϕ|ϕ dx +a²ε²

|x|^a−2

w_ε^μ2b+2

ϕ²dx+2C1(b+1)

|x|^a+γ∞

w_ε^μp+2b+1

ϕ²dx.

Then, we see from Caffarelli–Kohn–Nirenberg inequality [12] that for someC2>0,depend only onn, aandb,

|x|^2an/(n−2)w_ε^μb+1ϕ^2n/(n−2)dx

(n−2)/2

C₂

|x|^a

w_ε^μ2b+2

|∇ϕ|²+a|x|^a−1

w^μ_ε2b+2

|∇ϕ|ϕ dx +C₂a²

|x|^a−2

w^μ_ε2b+2

ϕ²dx+C₂b+1 ε²

|x|^a+γ∞

w_ε^μp+2b+1

ϕ²dx. (20)

Suppose that supp(ϕ)⊂B(y,1)⊂Rⁿ\B(0, R0).From Lemma 4 and the constraint

Rⁿ

χ_ε^μ w^μ_εβ˜

dx η^μ_ε^p_p−1⁺¹

,