Existence of standing wave solutions for coupled quasilinear Schrödinger systems with

Existence of standing wave solutions for coupled quasilinear Schrödinger systems with

critical exponents in R ^N

Li-Li Wang

, Xiang-Dong Fang

and Zhi-Qing Han

1School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, PR China

2School of Mathematics, Tonghua Normal Uninversity, Tonghua 134002, Jilin, PR China

3State Key Laboratory of Structural Analysis for Industrial Equipment Department of Engineering Mechanics, Dalian University of Technology

Received 10 June 2016, appeared 28 February 2017 Communicated by Dimitri Mugnai

Abstract. This paper is concerned with the following quasilinear Schrödinger system inR^N: (

−ε²∆u+V1(x)u−ε²∆(u²)u=K1(x)|u|^22∗−2u+h1(x,u,v)u,

−ε²∆v+V₂(x)v−ε²∆(v²)v=K₂(x)|v|^22∗−2v+h₂(x,u,v)v,

where N ≥ 3, V_i(x) is a nonnegative potential, K_i(x) is a bounded positive function, i = 1, 2. h₁(x,u,v)u and h2(x,u,v)v are superlinear but subcritical functions. Under some proper conditions, minimax methods are employed to establish the existence of standing wave solutions for this system provided thatεis small enough, more precisely, for any m ∈ N, it has mpairs of solutions if εis small enough. And these solutions (_uε,vε) →(_{0, 0})in some Sobolev space asε→0. Moreover, we establish the existence of positive solutions whenε=1. The system studied here can model some interaction phenomena in plasma physics.

Keywords: quasilinear Schrödinger system, critical growth, standing wave solutions, mountain pass theorem,(PS)_csequence.

2010 Mathematics Subject Classification: 35J50, 35J60, 35Q55.

1 Introduction

In this article we discuss the following coupled quasilinear Schrödinger system with critical exponents inR^N

(−ε²∆u+V₁(x)u−ε²∆(u²)u=K₁(x)|u|^22∗−2u+h₁(x,u,v)u,

−ε²∆v+V₂(x)v−ε²∆(v²)v=K₂(x)|v|^22∗−2v+h₂(x,u,v)v. (1.1)

BCorresponding author. Email: hanzhiq@dlut.edu.cn.

In recent years, much attention has been devoted to the quasilinear Schrödinger equation of the form:

−ε²∆u+V(x)u−ε²∆(u²)u=h(x,u), (1.2) whereε>0 is a small parameter (e.g. see [28,31]). Part of the interest is due to the fact that the solution of(1.2)is closely related to the existence of solitary wave solutions for the following equation:

iε∂_tw= −ε²∆w+V(x)w− f(|w|²)w−ε²k∆h(|w|²)h⁰(|w|²)w, (1.3) wherew : R×_R^N → _C,V(x)is a given potential, k is a real constant, f,h are suitable func- tions. In fact, the quasilinear equation (_1.3) has been derived as models of several physical phenomena. For example, it models the superfluid film equation in plasma physics [20], in self-channeling of a high-power ultra short laser in matter [3,6,24], in condensed matter the- ory [22] etc. It is worth pointing out that the related semilinear Schrödinger equation arises in many mathematical physics problems and has been extensively studied. We only mention [9,11,19,23] and the references therein. Also, there are more and more papers being con- cerned with semilinear Schrödinger system involving two condensate amplitudesw₁,w2. For example, Chen and Zhou [7] proved the uniqueness of positive solutions under some condi- tions for a coupled Schrödinger system. Tang [27] was concerned with multi-peak solutions to coupled Schrödinger systems with Neumann boundary conditions in a bounded domain ofR^N for N = 2, 3 and proved that all peaks locate either near the local maxima or near the local minima of the mean curvature at the boundary of the domain. Yang, Wei and Ding [30]

studied a Schrödinger system with nonlocal nonlineatities of Hartree type. Ye and Peng [32]

considered a coupled Schrödinger system with doubly critical exponents on R^N, which can be seen as a counterpart of the Brezis–Nirenberg problem.

Recently quasilinear systems also have been the focus for some researchers (e.g. [16,17,25]).

But compared with semilinear systems, only a few papers are known for them. Guo and Tang [17] proved the existence of a ground state solution by using Nehari manifold and concentra- tion compactness principle in a Orlicz space. Severo and Silva [25] established the existence of standing wave solutions for quasilinear Schrödinger systems involving subcritical nonlin- earities in Orlicz spaces. By referring to some arguments and methods in [11,25,30,31], we consider the quasilinear Schrödinger systems(1.1)with critical nonlinearities and discuss the existence of a positive solution and multiple solutions asεis small. Of particular interest to our paper is the results in [31], where the authors investigated the quasilinear Schrödinger equa- tion(1.2)with critical exponent h(x,u) = K(x)|u|^22∗−2+Hu(x,u) and proved it has at least one positive solution and multiple solutions whenε is small,where H_u(x,u)is a superlinear but subcritical function and satisfies some suitable conditions. The difficulty is caused by the usual lack of compactness since these problems involve critical exponents and are dealt with in the whole R^N. We remark that most papers above use the Cerami condition. But in this paper we prove that(PS)_c condition also holds. We suppose that the following assumptions are satisfied, wherei=1, 2:

(V₁) V_i ∈ C(_R^N,R) and there is a constant b > 0 such that m{x ∈ _R^N : V_i(x) < b} < _∞, where m denotes the Lebesgue measure;

(V₂) 0=V_i(0)≤V_i(x)≤maxV_i < +_∞;

(K) 0< C≤K_i ∈C(_R^N,R)∩L^∞(_R^N).

The functionsh₁,h₂ ∈C(_R^N×_R×_R,R⁺)and satisfy the following conditions.

(H₁) There is a constant 4 < µ< 22^∗ satisfyingµH(x,u,v) ≤ h₁(x,u,v)u²+h₂(x,u,v)v² for all(x,u,v)∈_R^N×_R×_{R, where} H(x,u,v) =Ru

0 h₁(x,t,v)tdt=Rv

0 h2(x,u,t)tdt.

(H₂) h₁(x,u,v)u = o(|(u,v)|) andh₂(x,u,v)v = o(|(u,v)|) uniformly in x ∈ _R^N as (u,v)→ (_{0, 0})_.

(H₃) There exist constants C₁,C₂ > 0 and p ∈ [3, 22^∗ −1) such that |h₁(x,u,v)u|+

|h₂(x,u,v)v| ≤C₁+C₂|(u,v)|^p−1 for all(x,u,v)∈ _R^N×_R×_R.

(H₄) H(x,u,v)≥C[|(u,v)|²+|(u,v)|]^q, where q∈(2, 2^∗)is a constant.

(H₅) h₁(x,−u,v) =h₁(x,u,v)andh₂(x,u,−v) =h₂(x,u,v)for all(x,u,v)∈ _R^N×_R×_R.

Notations. We collect below a list of the main notation used throughout this paper.

• Cwill denote various positive constants whose value may change from line to line.

• If the functions f and g satisfy ^f

(x) g(x)

≤ C, x ∈ U⁰(x₀), then we define f(x) = O(g(x)) as x→x₀.

• |u|denote the Euclidean norm ofu∈_R².

• The domain of integration isR^N by default.

• R

f(x)dxwill be represented by R f(x).

• We use L^s(_R^N), 1≤s≤ ∞, to denote the usual Lebesgue spaces with the norms

|u|_s := _Z

|u|^{s 1}_s

, 1≤ s<_∞,

kuk_∞ :=inf{C>0 :|u(x)| ≤Calmost everywhere inR^N}.

• Sdenotes the best Sobolev constant for H¹(_R^N)_.

Theorem 1.1. Assume that (V₁)–(V₂),(K)and(H₁)–(H₅)are satisfied. Then for anyσ > 0, there isτ_σ >0such that if ε ≤τ_σ, system(1.1)has at least one positive solutionu_ε = (u_ε,v_ε). Moreover, for any m ∈_Nandσ >0, there isτσm > 0such that if ε≤ τσm, system (1.1)has at least m pairs of solutionsuε = (uε,vε)→(0, 0)inEasε→0, whereEis stated later, satisfying

µ−4 2µ

Z

[ε²(₁+2u²_ε)|∇u_ε|²+V₁(x)u²_ε +ε²(₁+2v²_ε)|∇v_ε|²+V₂(x)v²_ε]≤σε^N

and 1

2N Z

[K₁(x)|uε|^22∗+K₂(x)|vε|^22∗] +^µ−4 4

Z

H(x,uε,vε)≤σε^N.

The existence and multiplicity of solutions for system(1.1)depends on the small param- eter ε. If the parameterεis not small enough, such asε ≡1, we cannot get the similar results as Theorem1.1unless we add some suitable conditions, wherei=1, 2:

(V₃) V_i ∈ C(_R^N_,R)is 1-periodic in x_j, 1 ≤ j ≤ N, and there is a constant a₀ > 0 such that V_i(x)≥a₀>0, ∀x∈ _R^N.

(K⁰) K_i ∈C(_R^N_,R)is 1-periodic in x_j, 1≤j≤ N, and there is a pointx₀∈_R^N _{such that}

(i) K_i(x₀) =sup_x∈RNK_i(x)>0.

(ii) K_i(x) =K_i(x₀) +O(|x−x₀|²), asx→x₀.

The functionsh1,h2 ∈C(_R^N×_R×_R,R⁺)and satisfy(_H1)_–(_H4)_and (H₆) h₁,h₂ is 1-periodic inx_j, 1≤ j≤ N.

Theorem 1.2. Letε = 1. Assume that (V₃), (K⁰), (H₁)–(H₄) and(H₆)are satisfied. Then system (_1.1)has at least one positive solutionu= (u,v)if N and q satisfy one of the following two conditions:

(N₁) 3≤ N<6and ^N_N⁺₋²₂ <q<2^∗; (N2) N≥6and2<q<2^∗.

Remark 1.1. Guo and Li in [18] discussed a class of modified nonlinear Schrödinger systems (Σ_i,j^N₌₁D_j(a_ij(u)D_iu)−¹_2Σ^N_i,j=1Dsa_ij(u)D_iuD_ju−a(x)u+Fu(u,v) =0,

Σ_i,j^N₌₁D_j(a_ij(v)D_iv)− ¹_2Σ^N_i,j=1D_sa_ij(v)D_ivD_jv−a(x)v+F_v(u,v) =0, (1.4) where F(u,v) = |u|^α|v|^β+|u|^p|v|^q, α,β,p,q > 1, α+β = 22^∗ and4 < p+q < 22^∗, and they proved the existence of a ground state positive solution by using a perturbation method. For the special case of a_ij(s) = (1+2s²)δ_ij, system(1.4)can be rewritten as

−_∆uj+V_j(x)u_j−_∆(u²_j)u_j = _Σ²_i6=jβ_ij(|u_i|^αi|u_j|^βj +|u_i|^pi|u_j|^qj), j=1, 2. (1.5) Comparing with(1.5), the coupling term in the present paper is not critical growth, but is more general than the coupling subcritical term of (1.5). The subcritical nonlinearities of (1.5) do not satisfy our condition(H₄). Hence, the proof in this paper is different from the one in [18].

The organization of this paper is as follows. In Section 2, we introduce the variational framework and restate the problem in a equivalent form by replacingε⁻²withλ. Furthermore, we reduce the quasilinear problem into a semilinear one by making change of variables and show some preliminary results. In Section 3, we prove the behaviors of the bounded (PS)_c sequences and then show that the energy functional satisfies the(PS)_ccondition under some suitable conditions. In Section 4, we verify the geometry of the mountain pass theorem and estimate the minimax values. In Section 5, we complete the proof of Theorem1.1. In the final section, we prove Theorem1.2.

2 An equivalent variational problem

To prove the existence of standing wave solutions of system(1.1)for smallε, we rewrite (1.1) in a equivalent form. Letλ=ε⁻². Then system(1.1)can be rewritten as

(−_∆u+λV₁(x)u−_∆(u²)u =λK₁(x)|u|^22∗−2u+λh₁(x,u,v)u,

−_∆v+λV₂(x)v−_∆(v²)v=λK₂(x)|v|^22∗−2v+λh₂(x,u,v)v, (2.1) forλ→+_∞.

We introduce the Hilbert spaces E_i :=

u∈ H¹(_R^N): Z

V_i(x)u²< _∞

with inner products

(u,v)_i :=

Z

∇u∇v+V_i(x)uv and the associated norms

kuk²_i = (u,u)_i, i=1, 2.

We shall work in the product space E = E₁×E₂ with elements u = (u,v). Thus, the norm in E can be defined as kuk² = kuk²₁+kvk²₂. It follows from (V₁) and (V2) that E_i embeds continuously inH¹(_R^N)(e.g. see [12]) and consequentlyEembeds continuously inH¹(_R^N)× H¹(_R^N). Notice that the normk · k_i is equivalent tok · k_i,λ induced by the inner product

(u,v)_i,λ :=

Z

∇u∇v+λV_i(x)uv

for each λ>0. Hencek · kis equivalent to the normk · k_λ induced by (u,v)_λ :=

Z

∇u₁∇v₁+λV₁(x)u₁v₁+

Z

∇u₂∇v₂+λV₂(x)u₂v₂.

It is thus clear that, for each s ∈ [2, 2^∗], there is a νs > 0 being independent of λ such that if λ≥1

|_u|_s ≤ν_sk_uk ≤ν_sk_uk_λ, ∀_u∈ _E, where| · |_s denotes the standard norm inL^s(_R^N)×L^s(_R^N)_.

Associated to system(2.1), the energy functional is J(u₁):= ¹

2 Z

(1+2u²₁)|∇u₁|²+λV₁(x)u²₁+ (1+2v²₁)|∇v₁|²+λV₂(x)v²₁

− ^λ 22^∗

Z

K₁(x)|u₁|^22∗+K₂(x)|v₁|^22∗−λ Z

H(x,u₁,v₁),

which is not well defined in H¹(_R^N)×H¹(_R^N). To save from this trouble, we make use of a change of variables u:= f⁻¹(u₁),v := f⁻¹(v₁)(see [8,10,13,21]), where f is defined by

f⁰(t) = _p ¹

1+2f²(t) ^on [0,+_∞) and f(t) =−f(−t) on(−_{∞, 0}]. We list some properties of f. Their proofs may be found in the above references.

Lemma 2.1. The function f satisfies the following properties:

(i) f is uniquely defined, C^∞ and invertible;

(ii) |f⁰(t)| ≤1for all t∈_R;

(iii) |f(t)| ≤ |t|for all t∈_R;

(iv) f(t)/t →1as t→0;

(v) f(t)/√

t→2^1/4 as t→+_∞;

(vi) f(_t)_/2≤t f⁰(_t)≤ f(_t)_{for t}≥0;

(vii) |f(t)| ≤₂^1/4|t|^1/2 for all t∈ _R;

(viii) there exists a positive constant C such that |f(t)| ≥ C|t|for |t| ≤ 1and |f(t)| ≥ C|t|^1/2 for

|t| ≥1;

(ix) |f(t)f⁰(t)|<1/√

2for all t∈_R;

(x) there exists a positive constant A such that

f^22∗(t) =2^NN−2t^2∗−At^2∗−1lnt+O(t^2∗−1), as t→+_∞.

After the change of variables, we obtain the following functional Φλ(_u):= ¹

2 Z

|∇u|²+λV₁(x)f²(u) +|∇v|²+λV₂(x)f²(v)

− ^λ 22^∗

Z

K₁(x)|f(u)|^22∗+K₂(x)|f(v)|^22∗−λ Z

H(x,f(u),f(v)).

ThenΦλ is well-defined onEand belongs toC¹ under hypotheses(V₁),(V₂), (K)and(H₃). Furthermore, we can check that

h_Φ⁰_λ(u),wi=h_Φ⁰_λ(u,v),(ϕ,ψ)i

=

Z

∇u∇ϕ+λV₁(x)f(u)f⁰(u)ϕ+∇v∇ψ+λV2(x)f(v)f⁰(v)ψ

−λ Z

K₁(x)|f(u)|^22∗−2f(u)f⁰(u)ϕ+K₂(x)|f(v)|^22∗−2f(v)f⁰(v)ψ

−λ Z

h₁(x,f(u),f(v))f(u)f⁰(u)ϕ+h₂(x,f(u),f(v))f(v)f⁰(v)ψ,

for allu,w∈E. We observe that ifu= (u,v)∈_Eis a critical point of the functionalΦλ, then it is a weak solution of the following system associated with the functionalΦλ

(−_∆u+λV₁(x)f(u)f⁰(u) =λK₁(x)|f(u)|^22∗−2f(u)f⁰(u) +λh₁(x,f(u),f(v))f(u)f⁰(u),

−_∆v+λV₂(x)f(v)f⁰(v) =λK₂(x)|f(v)|^22∗−2f(v)f⁰(v) +λh₂(x,f(u)_, f(v))f(v)f⁰(v)_. (2.2) Hence(f(u),f(v))is a weak solution of system(2.1)(cf. [8]). Theorem1.1can be restated as Theorem 2.1. Assume that(V₁)–(V₂),(K)and(H₁)–(H₅)are satisfied. Then for anyσ>0, there is Λσ>0such that ifλ≥_Λσ, system(2.2)has at least one positive solutionu_λ = (u_λ,v_λ). Moreover, for any m∈ _Nandσ >0, there isΛσm >₀such that if λ≥ _Λσm, system(_2.2)has at least m pairs of solutionsu_λ = (u_λ,v_λ), converging to(0, 0)inEasλ→_∞and satisfying

1 2N

Z

K₁(x)|f(u_λ)|^22∗+K₂(x)|f(v_λ)|^22∗+ ^µ−4 4

Z

H(x,f(u_λ),f(v_λ))≤σλ⁻

N 2

and

µ−4 2µ

Z

|∇u_λ|²+λV₁(x)f²(u_λ) +|∇v_λ|²+λV₂(x)f²(v_λ)≤σλ¹⁻

N 2. Remark 2.1. In order to get the positive solution, we introduce

Φ⁺_λ(u):= ¹ 2

Z

|∇u|²+λV₁(x)f²(u) +|∇v|²+λV₂(x)f²(v)

− ^λ 22^∗

Z

K₁(x)|f(u⁺)|^22∗+K₂(x)|f(v⁺)|^22∗−λ Z

H(x,f(u⁺),f(v⁺)), whereu⁺ :=max{u, 0},v⁺ :=max{v, 0}. ThenΦ⁺_λ ∈ C¹and the critical points ofΦ⁺_λ are the positive solutions of system (2.2).

3 Behavior of ( PS )

sequences

At this point, we recall that a sequence(un)⊂_Eis a(PS)_csequence at level c ((PS)_csequence for short), if Φλ(u_n) → c andΦ⁰_λ(u_n) → 0. Φλ is said to satisfy the (PS)_c condition if any (PS)_c sequence contains a convergent subsequence. However, due to the unboundedness of the domain and the critical term, we can not prove the (PS)_c condition holds in general. By establishing several lemmas, we will discuss the behaviors of(PS)_c sequences.

Lemma 3.1. Suppose that(V₂),(K)and(H₁)hold. Let(_un)⊂_Ebe a(PS)_csequence forΦλ. Then c≥0and(un)is bounded inE.

Proof. Set(u_n)to be a(PS)_c sequence:

Φλ(_un)→c, Φ⁰_λ(_un)→_0, n→_∞.

By Lemma2.1(vi) and(H₁), one sees that c+o(1) +o(1)ku_nk_λ ≥_Φλ(u_n)− ²

µh_Φ⁰_λ(u_n),u_ni

≥ 1

2− ² µ

Z

|∇un|²+λV₁(x)f²(un) +|∇vn|²+λV2(x)f²(vn)

− 1

22^∗ − ¹ µ

λ

Z

K₁(x)|f(u_n)|^22∗+K₂(x)|f(v_n)|^22∗. (3.1) Hence

(R

|∇u_n|²+λV₁(x)f²(u_n) +|∇v_n|²+λV₂(x)f²(v_n)≤ c+o(₁) +o(₁)k_unk_λ,

R K₁(x)|f(u_n)|^22∗+K₂(x)|f(v_n)|^22∗ ≤c+o(1) +o(1)ku_nk_λ. (3.2) From (3.2), we only need to prove that λR

V₁(x)|u_n|²+V₂(x)|v_n|² ≤ c+o(1) +o(1)k_unk_λ. We write that

λ Z

V₁(x)|u_n|²=λ Z

|un|≥1V₁(x)|u_n|²dx+λ Z

|un|≤1V₁(x)|u_n|²dx.

Combining(V₂),(K), (3.2) and Lemma2.1(viii), we have λ

Z

|un|≥1V₁(x)|un|²dx≤CλmaxV₁ Z

|un|≥1

|f(un)|^22∗dx

≤C Z

|un|≥1K₁(x)|f(u_n)|^22∗dx

≤c+o(1) +o(1)ku_nk_λ and

λ Z

|un|≤1V₁(x)|un|²dx ≤ ^λ C²

Z

|un|≤1V₁(x)f²(un)dx

≤c+o(₁) +o(₁)k_unk_λ. Thus λR

V₁(x)|un|² ≤c+o(1) +o(1)kunk_λ. Similarly, we can getλR

V₂(x)|vn|²≤ c+o(1) + o(1)ku_nk_λ. Thenku_nk²_λ ≤ c+o(1) +o(1)ku_nk_λ. Thus(u_n)is bounded inE. Taking the limit in (_3.1)we shows thatc≥_0.

By the above lemma, we know that every (PS)_c sequence (u_n) is bounded. We may assume up to a subsequence that un * u in E and in L^s×L^s, 2 ≤ s ≤ 2^∗, un → u in L^s_loc×L^s_loc, 1≤s <2^∗ andu_n(x)→u(x)a.e. inR^N. Clearlyuis a critical point of Φλ.

Lemma 3.2. Let(_un)be stated as in Lemma3.1 and s ∈ [2, 2^∗). There is a subsequence(_unj)such that for eache>0, there exists Re >0with

lim sup

j→_∞ Z

B_j\BR

|u_nj|^sdx≤e and

lim sup

j→_∞ Z

Bj\BR

|v_nj|^sdx≤e for all R≥R_e.

Proof. The proof is similar as that in [11]. We omit it here.

For notational convenience, we can assume in the following that Lemma3.2holds for both s = 2 ands = ^p+₂¹ with the same subsequence. Let η : [0,∞) → [0, 1]be a smooth function satisfyingη(t) =1 ift≤1,η(t) =0 ift ≥2. Define ˜u_j(x) =η ^2|_j^x|

u(x). It is known that

k_u−u˜jk_λ →0, as j→_∞. (3.3)

Lemma 3.3. Let(un_j)be stated as in Lemma3.2. Then

jlim→_∞ Z

|f(un_j)|^p− |f(un_j−u˜_j)|^p− |f(u˜_j)|^p=0 and

jlim→_∞ Z

|f(v_nj)|^p− |f(v_nj−v˜_j)|^p− |f(v˜_j)|^p=_0, where p∈ [2, 22^∗].

Proof. We only show that the first equality holds. As in [29], for any fixede> 0, there exists Ce>0 such that, for alla,b∈_R

||a+b|^q− |a|^q| ≤e|a|^q+C_e|b|^q, 1≤ q<+_∞.

We deduce that, by Lemma2.1(ix), for any fixede>0, there existsC_e>0 such that

|f(un_j)|^p− |f(un_j −u˜_j)|^p= |f²(un_j)|^p2 − |f²(un_j −u˜_j)|^p2

≤ e|f²(u_nj −u˜_j)|^p2 +Ce|f²(u_nj)− f²(u_nj −u˜_j)|^p2

≤ e|f(u_nj−u˜_j)|^p+C_e|₂f(u_nj−θu˜_j)f⁰(u_nj −θu˜_j)u˜_j|^2p

≤ e|f(u_nj−u˜_j)|^p+Ce|u˜_j|^p2, where and belowθ ∈(0, 1). Then by Lemma2.1(vii)

Γ^e_nj := (|f(u_nj)|^p− |f(u_nj−u˜_j)|^p− |f(u˜_j)|^p−e|f(u_nj−u˜_j)|^p)⁺

≤ |f(u˜_j)|^p+C_e|u˜_j|^p2

≤C_e|u|^2p.

The Lebesgue Dominated Convergence Theorem implies thatR _Γe

n_j →0 as j→_{∞. Hence}

Z

|f(u_nj)|^p− |f(u_nj −u˜_j)|^p− |f(u˜_j)|^p ≤

Z

Γ^e_nj+e|f(u_nj −u˜_j)|^p≤ Ce.

Lemma 3.4. Let(un_j)be stated as in Lemma3.2. Denote by

h_nj(x):=h₁(x,f(u_nj),f(v_nj))f(u_nj)f⁰(u_nj)−h₁(x,f(u˜_nj),f(v˜_nj))f(u˜_nj)f⁰(u˜_nj)

−h₁(x,f(u_nj −u˜_j),f(v_nj−v˜_j))f(u_nj−u˜_j)f⁰(u_nj−u˜_j) and

g_nj(x):=h₂(x,f(u_nj),f(v_nj))f(v_nj)f⁰(v_nj)−h₂(x,f(u˜_nj),f(v˜_nj))f(v˜_nj)f⁰(v˜_nj)

−h₂(x,f(u_nj −u˜_j),f(v_nj−v˜_j))f(v_nj −v˜_j)f⁰(v_nj−v˜_j). We have

jlim→_∞ Z

hn_j(x)ϕ=0 and

jlim→_∞ Z

g_nj(x)ψ=0 uniformly forkwk_λ= k(ϕ,ψ)k_λ ≤1.

Proof. Note that (3.3) and the local compactness of the Sobolev embedding theorem imply that, for anyR>0

jlim→_∞

Z

BR

h_nj(x)ϕdx

=_{0 (3.4)}

uniformly forkϕk_1,λ ≤1. For anyε>0, from (3.3) and the integrability of|u|^sonR^N, we can choose R>0 such that

lim sup

j→_∞ Z

Bj\BR

|u˜_j|^sdx≤

Z

B^c_R

|u|^sdx≤ε.

Combining(H₂),(H₃)and Lemma2.1(ii), (iii), (vii), we get that

|h₁(x,f(u),f(v))f(u)f⁰(u)ϕ| ≤C[|(f(u),f(v))|+|(f(u),f(v))|^p−1]|ϕ|

≤C(|u||ϕ|+|v||ϕ|+|u|^p−21|ϕ|+|v|^p−21|ϕ|). (3.5) Therefore, it follows from (3.4), (3.5), the Hölder inequality and Lemma3.2 that

lim sup

j→_∞

Z

hn_j(x)ϕ

≤ lim sup

j→_∞ Z

B_j\BR

|h_nj(x)ϕ|dx

≤Clim sup

j→_∞ Z

B_j\B_R

(|u_nj|+|u_nj −u˜_j|+|u˜_j|)|ϕ|

+ (|v_nj|+|v_nj −v˜_j|+|v˜_j|)|ϕ|+|u_nj|^p−21 +|u_nj−u˜_j|^p−21 +|u˜_j|^p−21|ϕ| +|v_nj|^p−21 +|v_nj −v˜_j|^p−21 +|v˜_j|^p−21|ϕ|

≤Clim sup

j→_∞ Z

Bj\BR

|un_j|²dx ¹₂

+ _Z

Bj\BR

|u˜_j|²dx ¹₂

+ Z

Bj\BR

|vn_j|²dx ¹₂

+ Z

Bj\BR

|v˜_j|²dx

¹₂ Z

|ϕ|^{2 1}₂

+Clim sup

j→_∞ Z

Bj\BR

|u_nj|^p+21dx ^p_p⁻₊¹₁

+ _Z

Bj\BR

|u˜_j|^p+21dx ^p_p⁻₊¹₁

+ _Z

B_j\B_R

|v_nj|^p+21dx ^p_p⁻₊¹₁

+ _Z

B_j\B_R

|v˜_j|^p+21dx

^p_p⁻₊¹_{1 Z}

|ϕ|^p+21dx _p+²₁

≤Ce¹² +Ce^p

−1 p+1

uniformly forkϕk_1,λ ≤1. Similarly, we can get that the other equality holds.

Lemma 3.5. Let(un_j)be stated as in Lemma3.2. One has along a subsequence:

(i) Φλ(u_nj−u˜_j)→c−_Φλ(u). (ii) Φ⁰_λ(un_j−u˜_j)→0.

Proof. (i)Obviously, we can see Φλ(u_nj−u˜_j) =_Φλ(u_nj)−_Φλ(u˜_j)− ^λ

2 Z

V₁(x)[|f(u_nj)|²− |f(u_nj −u˜_j)|²− |f(u˜_j)|²]

−^λ 2

Z

V₂(x)[|f(v_nj)|²− |f(v_nj−v˜_j)|²− |f(v˜_j)|²] + ^λ

22^∗ Z

K₁(x)[|f(u_nj)|^22∗− |f(u_nj−u˜_j)|^22∗− |f(u˜_j)|^22∗] + ^λ

22^∗ Z

K₂(x)[|f(v_nj)|^22∗− |f(v_nj−v˜_j)|^22∗− |f(v˜_j)|^22∗] +λ

Z

H(x,f(un_j),f(vn_j))−H(x,f(un_j −u˜_j),f(vn_j−v˜_j))−H(x,f(u˜_j),f(v˜_j)). We claim that

jlim→_∞ Z

V₁(x)[|f(u_nj)|²− |f(u_nj −u˜_j)|²− |f(u˜_j)|²] =0, (3.6)

jlim→_∞ Z

V₂(x)[|f(v_nj)|²− |f(v_nj−v˜_j)|²− |f(v˜_j)|²] =0, (3.7)

jlim→_∞ Z

K₁(x)[|f(un_j)|^22∗− |f(un_j −u˜_j)|^22∗− |f(u˜_j)|^22∗] =0, (3.8)

jlim→_∞ Z

K₂(x)[|f(v_nj)|^22∗− |f(v_nj −v˜_j)|^22∗− |f(v˜_j)|^22∗] =0, (3.9)

jlim→_∞ Z

H(x,f(u_nj),f(v_nj))−H(x,f(u_nj−u˜_j),f(v_nj −v˜_j))−H(x,f(u˜_j), f(v˜_j)) =0. (3.10) By conditions (V₂), (K) and Lemma 3.3, we conclude that (3.6)–(3.9) hold. Similar to the proof of Lemma 3.4, it is easy to see that (3.10) holds. Using the fact Φλ(u_nj) → c and Φλ(_u˜j)→_Φλ(_u), we get conclusion 1.

(ii)We first notice that, for any given w= (ϕ,ψ)∈_Esatisyingkwk_λ ≤1, (_Φ⁰_λ(un_j−u˜_j),w) = (_Φ⁰_λ(un_j),w)−(_Φ⁰_λ(u˜_j),w)

−λ Z

V₁(x)[f(u_nj)f⁰(u_nj)− f(u_nj −u˜_j)f⁰(u_nj −u˜_j)− f(u˜_j)f⁰(u˜_j)]ϕ

−λ Z

V₂(x)[f(v_nj)f⁰(v_nj)− f(v_nj −v˜_j)f⁰(v_nj−v˜_j)− f(v˜_j)f⁰(v˜_j)]ψ +λ

Z

K₁(x)^h|f(un_j)|^22∗−2f(un_j)f⁰(un_j)

− |f(_unj−u˜_j)|^22∗−2f(_unj−u˜_j)_f⁰(_unj−u˜_j)− |f(u˜_j)|^22∗−2f(u˜_j)_f⁰(u˜_j)ⁱϕ +λ

Z

K₂(x)^h|f(v_nj)|^22∗−2f(v_nj)f⁰(v_nj)

− |f(v_nj−v˜_j)|^22∗−2f(v_nj−v˜_j)f⁰(v_nj −v˜_j)− |f(v˜_j)|^22∗−2f(v˜_j)f⁰(v˜_j)ⁱψ +λ

Z

hn_jϕ+λ Z

gn_jψ,

whereh_nj(x)_andg_nj(x)are stated in Lemma3.4. Noticing the boundedness of(_unj)_{inE, the} equality

d|f(t)|^22∗−2f(t)f⁰(t)

dt =C|f(t)|^22∗−2|f⁰(t)|²+|f(t)|^22∗−2f(t)f⁰⁰(t)

=C|f(t)|^22∗−2|f⁰(t)|²−2|f(t)|^22∗|f⁰(t)|⁴,

the mean value theorem, Lemma2.1(vii), (ix) and the Hölder inequality, we have forR>0 Z

B^c_R

|f(u_nj)|^22∗−2f(u_nj)f⁰(u_nj)− |f(u_nj −u˜_j)|^22∗−2f(u_nj−u˜_j)f⁰(u_nj−u˜_j) |ϕ|dx

≤C Z

B^c_R

h|f(u_nj−θu˜_j)|^22∗−2|f⁰(u_nj −θu˜_j)|²+|f(u_nj −θu˜_j)|^22∗|f⁰(u_nj −θu˜_j)|⁴ⁱ|u˜_j||ϕ|dx

≤C Z

B^c_R

|un_j −θu˜_j|^2∗−2|u˜_j||ϕ|dx

≤C _Z

|u_nj−θu˜_j|^2∗

^{2∗ −}_2∗² _Z

B^c_R

|u˜_j|^2∗dx ₂¹_{∗ Z}

|ϕ|^2∗ ₂¹_∗

≤C _Z

B^c_R

|u|^2∗dx ₂¹_∗

kϕk_1,λ.

We have also that Z

B^c_R

|f(u˜_j)|^22∗−2f(u˜_j)f⁰(u˜_j)

|ϕ|dx≤C Z

B^c_R

|u˜_j|^2∗−1|ϕ|dx

≤C _Z

B^c_R

|u˜_j|^2∗dx

^{2∗ −}_2∗¹ _Z

|ϕ|^2∗ ₂¹_∗

≤C Z

B^c_R

|u|^2∗dx ^{2∗ −}₂∗¹

kϕk_1,λ.