Existence of nontrivial solutions to perturbed p-Laplacian system in ℝ N involving critical nonlinearity

  • Huixing Zhang1Email author and

    Affiliated with

    • Wenbin Liu1

      Affiliated with

      Boundary Value Problems20122012:53

      DOI: 10.1186/1687-2770-2012-53

      Received: 29 September 2011

      Accepted: 4 May 2012

      Published: 4 May 2012

      Abstract

      We consider a p-Laplacian system with critical nonlinearity in ℝ N . Under the proper assumptions, we obtain the existence of nontrivial solutions to perturbed p-Laplacian system by using the variational approach.

      MR Subject Classification: 35B33; 35J60; 35J65.

      Keywords

      p-Laplacian system critical nonlinearity variational methods.

      1 Introduction

      This article is concerned with the existence of solutions to the following nonlinear perturbed p-Laplacian system
      { ε p Δ p u + V ( x ) | u | p 2 u = K ( x ) | u | p * 2 u + H u ( u , v ) , x N , ε p Δ p v + V ( x ) | v | p 2 v = K ( x ) | v | p * 2 v + H v ( u , v ) , x N , u ( x ) , v ( x ) > 0 , u ( x ) , v ( x ) 0 as | x | , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equ1_HTML.gif
      (1.1)

      where Δ p u = div(|∇u|p-2u) is the p-Laplacian operator, 1 < p < N and p* = Np/(N − p) is the critical exponent.

      Throughout the article, we will assume that:

      (V0) VC(ℝ N ), V (0) = inf V (x) = 0 and there exists b > 0 such that the set ν b := {x ∈ ℝ N : V (x) < b} has finite Lebesgue measure;

      (K0) K(x) ∈ C(ℝ N ), 0 < inf K ≤ sup K < ∞;

      (H1) HC1(ℝ2) and H s , H t = o(|s|p-1+ |t|p-1) as |s| + |t| → 0;

      (H2) there exist c > 0 and p < q < p* such that
      | H s ( s , t ) | , | H t ( s , t ) | c ( 1 + | s | q - 1 + | t | q - 1 ) ; http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equa_HTML.gif

      (H3) There are a0> 0, θ ∈ (p, p*) and α, β > p such that H(s, t) ≥ a0(|s| α + |t| β ) and 0 < θH(s, t) ≤ sH s + tH t .

      Under the above mentioned conditions, we will get the following result.

      Theorem 1. If (V0), (K0) and (H1)-(H3) hold, then for any σ > 0, there is ε σ > 0 such that if ε < ε σ , the problem (1.1) has at least one positive solution (u ε , v ε ) which satisfy
      θ - p p θ N ( ε p | u ε | p + ε p | v ε | p + V ( x ) | u ε | p + V ( x ) | v ε | p ) σ ε N . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equb_HTML.gif
      The scalar form of the problem (1.1) is as follows
      - ε p Δ p u + V ( x ) | u | p - 2 u = K ( x ) | u | p * - 2 u + h ( x , u ) , x N . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equ2_HTML.gif
      (1.2)

      The Equation (1.2) has been studied in many articles. The case p = 2 was investigated extensively under various hypotheses on the potential and the nonlinearity by many authors including Brézis and Nirenberg [1], Ambrosetti [2] and Guedda and Veron [3] (see also their references) in bounded domains. As far as unbounded domains are concerned, we recall the work by Benci and Cerami [4], Floer and Weistein [5], Oh [6], Clapp [7], Del Pino and Felmer [8], Cingolani and Lazzo [9], Ding and Lin [10]. Especially, in [10], the authors studied the Equation (1.2) in the case p = 2. In that article, they made the following assumptions:

      (A1) VC(ℝ N ), min V = 0 and there is b > 0 such that the set ν b := {x ∈ ℝ N : V (x) < b} has finite Lebesgue measure;

      (A2) K(x) ∈ C(ℝ N ), 0 < inf K ≤ sup K <

      (B1) hC(ℝ N × ℝ) and h(x, u) = o(|u|) uniformly in x as |u| → 0;

      (B2) there are c0> 0, q < 2* such that |h(x, u)| ≤ c0(1 + |u|q-1) for all (x, u);

      (B3) there are a0> 0, p > 2 and µ > 2 such that H(x, u) = a0|u| p and µH(x, u) ≤ h(x, u)u for all (x, u), where H ( x , u ) = 0 u h ( x , s ) d s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq1_HTML.gif.

      That article obtained the existence of at least one positive solution u ε of least energy if the assumptions (A1)-(A2) and (B1)- (B3) hold.

      For the Equation (1.2) in the case p ≠ 2, we recall some works. Garcia Azorero and Peral Alonso [11] considered (1.2) with ε ≤ 1, V (x) = µ, K(x) = 1, h(x, u) = 0 and proved that (1.2) has a solution if p2N and µ ∈ (0, λ1), where λ1 is the first eigenvalue of the p-Laplacian. In [12], Alves and Ding studied the same problem of [11] and obtained the multiplicity of positive solutions in bounded domain Ω ⊂ ℝ N . Moreover, Liu and Zheng [13] investigated (1.2) in ℝ N with ε = 1 and K(x) = 0. Under the sign-changing potential and subcritical p-superlinear nonlinearity, the authors got the existence result.

      Motivated by some results found in [10, 11, 13], a natural question arises whether existence of nontrivial solutions continues to hold for the p-Laplacian system with the critical nonlinearity in ℝ N .

      The main difficulty in the case above mentioned is the lack of compactness of the energy functional associated to the system (1.1) because of unbounded domain ℝ N and critical nonlinearity. To overcome this difficulty, we make careful estimates and prove that there is a Palais-Smale sequence that has a strongly convergent sequence. The method or idea here is similar to the one of [10]. We can prove that the functional associated to (1.1) possesses (PS) c condition at some energy level c. Furthermore, we prove the existence result by using the mountain pass theorem due to Rabinowitz [14].

      The main result in the present article concentrates on the existence of positive solutions to the system (1.1) and can be seen as a complement of the results developed in [10, 11, 13].

      This article is organized as follows. In Section 2, we give the necessary notations and preliminaries. Section 3 is devoted to the behavior of (PS) c sequence and the mountain geometry structure. Finally, in Section 4, we prove the existence of nontrivial solution.

      2 Notations and preliminaries

      Let C 0 ( N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq2_HTML.gif denote the collection of smooth functions with compact support and D1,p(ℝ N ) be the completion of C 0 ( N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq3_HTML.gif under
      | | u | | p = N | u | p d x . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equc_HTML.gif
      We introduce the space
      E ( N , V ) = { u W 1 , p ( N ) : N V ( x ) | u | p < } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equd_HTML.gif
      equipped with the norm
      | | u | | E = N ( | u | p + V ( x ) | u | p ) 1 p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Eque_HTML.gif
      and the space
      E λ ( N , V ) = u W 1 , p ( N ) : N λ V ( x ) | u | p < , λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equf_HTML.gif
      under
      | | u | | λ = ( N | u | p + λ V ( x ) | u | p ) ) 1 p . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equg_HTML.gif

      Observe that ‖ · ‖ E is equivalent to the one ‖ · ‖ λ for each λ > 0. It follows from (V0) that E(ℝ N , V) continuously embeds in W1,p(ℝ N ).

      Set B = E λ × E λ and | | ( u , v ) | | λ = | | u | | λ p + | | v | | λ p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq4_HTML.gif for any (u, v) ∈ B. Let λ = ε-pin the system (1.1), then (1.1) is changed into
      { Δ p u + λ V ( x ) | u | p 2 u = λ K ( x ) | u | p * 2 u + λ H u ( u , v ) , N , Δ p v + λ V ( x ) | v | p 2 v = λ K ( x ) | v | p * 2 v + λ H v ( u , v ) , x N , u ( x ) , v ( x ) > 0 , u ( x ) , v ( x ) 0 , as | x | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equ3_HTML.gif
      (2.1)

      In order to prove Theorem 1, we only need to prove the following result.

      Theorem 2. Let (V0), (K0) and (H1)-(H3) be satisfied. Then for any σ > 0, there exists Λ σ > 0 such that if λ ≥ Λ σ , the system (2.1) has at least one least energy solution (u λ , v λ ) satisfying
      θ - p p θ N ( | u λ | p + | v λ | p + λ V ( x ) ( | u λ | p + | v λ | p ) ) σ λ 1 - N p . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equ4_HTML.gif
      (2.2)
      The energy functional associated with (2.1) is defined by
      I λ ( u , v ) = 1 p N ( | u | p + λ V ( x ) | u | p + | v | p + λ V ( x ) | v | p ) - λ p * N K ( x ) ( | u | p * + | v | p * ) - λ N H ( u , v ) = 1 p | | ( u , v ) | | λ p - λ N G ( u , v ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equh_HTML.gif

      where G ( u , v ) = 1 p * K ( x ) ( | u | p * + | v | p * ) + H ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq5_HTML.gif.

      From the assumptions of Theorem 2, standard arguments [14] show that I λ C1(B, ℝ) and its critical points are the weak solutions of (2.1).

      3 Technical lemmas

      In this section, we will recall and prove some lemmas which are crucial in the proof of the main result.

      Lemma 3.1. Let the assumptions of Theorem 2 be satisfied. If the sequence {(u n , v n )} ⊂ B is a (PS) c sequence for I λ , then we get that c ≥ 0 and {(u n , v n )} is bounded in the space B.

      Proof. One has
      I λ ( u n , v n ) - 1 θ I λ ' ( u n , v n ) ( u n , v n ) = 1 p | | ( u n , v n ) | | λ p - λ p * N K ( x ) ( | u n | p * + | v n | p * ) - λ N H ( u n , v n ) - 1 θ | | ( u n , v n ) | | λ p - λ N K ( x ) ( | u n | p * + | v n | p * ) - λ N ( u n H s ( u n , v n ) + v n H t ( u n , v n ) ) = 1 p - 1 θ | | ( u n , v n ) | | λ p + 1 θ - 1 p * λ N K ( x ) ( | u n | p * + | v n | p * ) + λ N 1 θ ( u n H s ( u n , v n ) + v n H t ( u n , v n ) ) - H ( u n , v n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equi_HTML.gif
      By the assumptions (K0) and (H3), we have
      I λ ( u n , v n ) - 1 θ I λ ( u n , v n ) ( u n , v n ) 1 p - 1 θ | | ( u n , v n ) | | λ p . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equj_HTML.gif

      Together with I λ (u n , v n ) → c and I λ ( u n , v n ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq6_HTML.gif as n → ∞, we easily obtain that the (PS) c sequence is bounded in B and the energy level c ≥ 0. □

      From Lemma 3.1, there exists (u, v) ∈ B such that (u n , v n ) ⇀ (u, v) in B. Furthermore, passing to a subsequence, we have u n u and v n v in L l o c d ( N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq7_HTML.gif for any d ∈ [p, p*) and u n u, v n v a.e. in ℝ N .

      Lemma 3.2. Let d ∈ [p, p*). There exists a subsequence { ( u n j , v n j ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq8_HTML.gif such that for any ε > 0, there is r ε > 0 with
      lim i sup B i \ B r ( | u n i | d + | v n i | d ) ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equk_HTML.gif

      for any r ≥ r ε , where B r := {x ∈ ℝ N : |x| ≤ r}.

      Proof. The proof of Lemma 3.2 is similar to the one of Lemma 3.2 of [10], so we omit it. □

      Let ηC(ℝ+) be a smooth function satisfying 0 ≤ η(t) 1, η(t) = 1 if t ≤ 1 and η(t) = 0 if t ≥ 2. Define ũ j ( x ) = η ( 2 | x | / j ) u ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq9_HTML.gif, j ( x ) = η ( 2 | x | / j ) v ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq10_HTML.gif. It is obvious that
      | | u - ũ j | | λ 0 and | | v - j | | λ 0 as  j . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equ5_HTML.gif
      (3.1)
      Lemma 3.3. One has
      lim j N ( H s ( u n j , v n j ) - H s ( u n j - ũ j , v n j - j ) - H s ( ũ j , j ) ) φ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equl_HTML.gif
      and
      lim j N ( H t ( u n j , v n j ) - H t ( u n j - ũ j , v n j - v j ) - H t ( ũ j , j ) ) ψ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equm_HTML.gif

      uniformly in (φ, ψ) ∈ B with ‖(φ, ψ B ≤ 1.

      Proof. From the assumptions (H1)-(H2) and Lemma 3.2, we have
      lim j sup N ( H s ( u n j , v n j ) H s ( u n j u ˜ j , v n j v ˜ j ) H s ( u ˜ j , v ˜ j ) ) φ = lim j sup B j ( H s ( u n j , v n j ) H s ( u n j u ˜ j , v n j v ˜ j ) H s ( u ˜ j , v ˜ j ) ) φ = lim j sup B j \ B r ( H s ( u n j , v n j ) H s ( u n j u ˜ j , v n j v ˜ j ) H s ( u ˜ j , v ˜ j ) ) φ c lim j sup B j \ B r ( | u n j | p 1 + | v n j | p 1 + | u n j | q 1 + | v n j | q 1 + | u ˜ j | p 1 + | v ˜ j | p 1 + | u ˜ j | q 1 + | v ˜ j | q 1 + | u n j u ˜ j | p 1 + | v n j v ˜ j | p 1 + | u n j u ˜ j | q 1 + | v n j v ˜ j | q 1 ) φ c 1 lim j sup B j \ B r ( | u n j | p 1 + | v n j | p 1 + | u ˜ j | p 1 + | v ˜ j | p 1 ) φ + c 2 lim j sup B j \ B r ( | u n j | q 1 + | v n j | q 1 + | u ˜ j | q 1 + | v ˜ j | q 1 ) φ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equ6_HTML.gif
      (3.2)
      By Hölder inequality and Lemma 3.2, it follows that
      lim j sup B j \ B r | u n j | p - 1 | φ | lim j sup B j \ B r | u n j | p p - 1 p B j \ B r | φ | p 1 p lim j sup B j \ B r | u n j | p p - 1 p N | φ | p 1 p lim j sup B j \ B r | u n j | p p - 1 p = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equn_HTML.gif
      and
      lim j sup B j \ B r | u n j | p - 1 | φ | lim j sup B j \ B r | u n j | p q - 1 p B j \ B r | φ | q 1 q lim j sup B j \ B r | u n j | q q - 1 q N | φ | q 1 q lim j sup B j \ B r | u n j | q q - 1 q = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equo_HTML.gif
      Similarly, we get
      lim j sup B j \ B r ( | v n j | p - 1 | + | ũ j | p - 1 + | j | p - 1 ) φ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equp_HTML.gif
      and
      lim j sup B j \ B r ( | v n j | q - 1 | + | ũ j | q - 1 + | j | q - 1 ) φ = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equq_HTML.gif
      Thus
      lim j N ( H s ( u n j , v n j ) - H s ( u n j - ũ j , v n j - j ) - H s ( ũ j , j ) ) φ = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equr_HTML.gif
      From the similar argument, we also get
      lim j N ( H t ( u n j , v n j ) - H t ( u n j - ũ j , v n j - j ) - H t ( ũ j , j ) ) ψ = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equs_HTML.gif

      Lemma 3.4. One has along a subsequence
      I λ ( u n - ũ n , v n - n ) c - I λ ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equt_HTML.gif
      and
      I λ ( u n - ũ n , v n - n ) 0 in  B - 1 ( the dual space of  B ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equu_HTML.gif
      Proof. From the Lemma 2.1 of [15] and the argument of [16], we have
      I λ ( u n - ũ n , v n - n ) = 1 p N ( | u n - ũ n | p + λ V ( x ) | u n - ũ n | p + | v n - n | p + λ V ( x ) | v n - n | p ) - λ p * N K ( x ) ( | u n - ũ n | p * + | v n - n | p * ) - λ N H ( u n - ũ n , v n - n ) = I λ ( u n , v n ) - I λ ( ũ n , n ) + λ p * N K ( x ) ( ( | u n | p * - | u n - ũ n | p * - | ũ n | p * ) + ( | v n | p * - | v n - n | p * - | n | p * ) ) + λ N ( H ( u n , v n ) - H ( u n - ũ n , v n - n ) - H ( ũ n , n ) ) + o ( 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equv_HTML.gif
      By (3.1) and the similar idea of proving the Brézis-Lieb Lemma [17], it is easy to get
      lim n N K ( x ) ( ( | u n | p * - | u n - ũ n | p * - | ũ n | p * ) + ( | v n | p * - | v n - n | p * - | n | p * ) ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equw_HTML.gif
      and
      lim n N ( H ( u n , v n ) - H ( u n - ũ n , v n n ) - H ( ũ n , n ) ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equx_HTML.gif
      In connection with the fact I λ (u n , v n ) → c and I λ ( ũ n , n ) I λ ( u , v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq11_HTML.gif, we obtain
      I λ ( u n - ũ n , v n - n ) c - I λ ( u , v ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equy_HTML.gif

      In the following, we will verify the fact I λ ( u n - ũ n , v n - n ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq12_HTML.gif.

      For any (φ, ψ) ∈ B, it follows that
      I λ ' ( u n u ˜ n , v n v ˜ n ) ( φ , ψ ) = I λ ' ( u n , v n ) ( φ , ψ ) I λ ' ( u ˜ n , v ˜ n ) ( φ , ψ ) + λ N K ( x ) [ ( | u n | p * 2 u n | u n u ˜ n | p * 2 ( u n u ˜ n ) | u ˜ n | p * 2 u ˜ n ) φ + ( | v n | p * 2 v n | v n v ˜ n | p * 2 ( v n v ˜ n ) | v ˜ n | p * 2 v ˜ n ) ψ ] + λ N [ ( H s ( u n , v n ) H s ( u n u ˜ n , v n v ˜ n ) H s ( u ˜ n , v ˜ n ) ) φ + ( H t ( u n , v n ) H t ( u n u ˜ n , v n v ˜ n ) H t ( u ˜ n , v ˜ n ) ) ψ ] + o ( 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equz_HTML.gif
      Standard argument shows that
      lim n N K ( x ) ( | u n | p * - 2 u n - | u n - ũ n | p * - 2 ( u n - ũ n ) - | ũ n | p * - 2 ũ n ) φ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equaa_HTML.gif
      and
      lim n N K ( x ) ( | v n | p * - 2 v n - | v n - n | p * - 2 ( v n - n ) - | n | p * - 2 n ) ψ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equab_HTML.gif

      uniformly in ‖φ, ψ)‖ B 1.

      By Lemma 3.3, we have
      lim n N ( H s ( u n , v n ) - H s ( u n - ũ n , v n - n ) - H s ( ũ n , n ) ) φ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equac_HTML.gif
      and
      lim n N ( H t ( u n , v n ) - H t ( u n - ũ n , v n - n ) - H t ( ũ n , n ) ) ψ = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equad_HTML.gif
      uniformly in ‖(φ, ψ)‖ B 1. From the facts above mentioned, we obtain
      I λ ( u n - ũ n , v n - n ) 0 in  B - 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equae_HTML.gif

      Let u n 1 = u n - ũ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq13_HTML.gif, v n 1 = v n - n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq14_HTML.gif, then u n - u = u n 1 + ( ũ n - u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq15_HTML.gif, v n - v = v n 1 + ( n - v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq16_HTML.gif. From (3.1), we get (u n , v n ) → (u, v) in B if and only if ( u n 1 , v n 1 ) ( 0 , 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq17_HTML.gif in B.

      Observe that
      I λ ( u n 1 , v n 1 ) - 1 p I λ ' ( u n 1 , v n 1 ) ( u n 1 , v n 1 ) = 1 p - 1 p * λ N K ( x ) ( | u n 1 | p * + | v n 1 | p * ) + λ N 1 p ( u n 1 H s ( u n 1 , v n 1 ) + v n 1 H t ( u n 1 , v n 1 ) ) - H ( u n 1 , v n 1 ) λ N N K ( x ) ( | u n 1 | p * + | v n 1 | p * ) λ N K min N ( | u n 1 | p * + | v n 1 | p * ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equaf_HTML.gif

      where K min = inf x N K ( x ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq18_HTML.gif.

      Thus by Lemma 3.4, we get
      | | ( u n 1 , v n 1 ) | | p * p * N ( c - I λ ( u , v ) ) λ K min + o ( 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equ7_HTML.gif
      (3.3)

      Now, we consider the energy level of the functional I λ below which the (PS) c condition hold.

      Let V b (x):= max{V (x), b}, where b is the positive constant in the assumption (V0). Since the set ν b has finite measure and u n 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq19_HTML.gif, v n 1 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq20_HTML.gif in L loc p ( N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq21_HTML.gif, we get
      N V ( x ) ( | u n 1 | p + | v n 1 | p ) = N V b ( x ) ( | u n 1 | p + | v n 1 | p ) + o ( 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equ8_HTML.gif
      (3.4)
      From (K0), (H1)-(H3) and Young inequality, there is C b > 0 such that
      N ( K ( x ) ( | u | p * + | v | p * ) + u H s ( u , v ) + v H t ( u , v ) ) b ( | | u | | p p + | | v | | p p ) + C b ( | | u | | p * p * + | | v | | p * p * ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equ9_HTML.gif
      (3.5)
      Let S be the best Sobolev constant of the immersion
      S | | u | | p * p N | u | p for all  u W 1 , p ( N ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equag_HTML.gif

      Lemma 3.5. Let the assumptions of Theorem 2 be satisfied. There exists α0> 0 independent of λ such that, for any (PS) c sequence {(u n , v n )} ⊂ B for I λ with (u n , v n ) ⇀ (u, v), either (u n , v n ) (u, v) or c - I λ ( u , v ) α 0 λ 1 - N p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq22_HTML.gif.

      Proof. Assume that (u n , v n ) ↛ (u, v), then
      lim inf n | | ( u n 1 , v n 1 ) | | λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equah_HTML.gif
      and
      c - J λ ( u , v ) > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equai_HTML.gif
      By the Sobolev inequality, (3.4) and (3.5), we get
      S ( | | u n 1 | | p * p + | | v n 1 | | p * p ) N ( | u n 1 | p + | v n 1 | p ) = N ( | u n 1 | p + λ V ( x ) | u n 1 | p + | v n 1 | p + λ V ( x ) | v n 1 | p ) - λ N V ( x ) ( | u n 1 | p + | v n 1 | p ) = λ N K ( x ) ( | u n 1 | p * + | v n 1 | p * ) + u n 1 H s ( u n 1 , v n 1 ) + v n 1 H t ( u n 1 , v n 1 ) - λ N V ( x ) ( | u n 1 | p + | v n 1 | p ) + o ( 1 ) λ b ( | | u n 1 | | p p + | | v n 1 | | p p ) + λ C b ( | | u n 1 | | p * p * + | | v n 1 | | p * p * ) - λ b ( | | u n 1 | | p p + | | v n 1 | | p p ) + o ( 1 ) = λ C b ( | | u n 1 | | p * p * + | | v n 1 | | p * p * ) + o ( 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equaj_HTML.gif
      This, together with lim inf n ( | | u n 1 | | p * p * + | | v n 1 | | p * p * ) > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq23_HTML.gif and (3.3), gives
      S λ C b ( | | u n 1 | | p * p * + | | v n 1 | | p * p * ) p * - p p * + o ( 1 ) λ C b N ( c - I λ ( u , v ) ) λ K min p N + o ( 1 ) = λ 1 - p N C b N K min p N ( c - I λ ( u , v ) ) p N + o ( 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equak_HTML.gif
      Set α 0 = S N p C b - N p N - 1 K min http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq24_HTML.gif, then
      α 0 λ 1 - N p c - I λ ( u , v ) + o ( 1 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equal_HTML.gif

      This proof is completed. □

      Since W 1 , p ( N ) L p * ( N ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq25_HTML.gif is not compact, I λ does not satisfy the (PS) c condition for all c > 0. But Lemma 3.5 shows that I λ satisfies the following local (PS) c condition.

      Lemma 3.6. From the assumptions of Theorem 2, there exists a constant α0> 0 independent of λ such that, if a (PS) c sequence {(u n , v n )} ⊂ B for I λ satisfies c α 0 λ 1 - N p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq26_HTML.gif, the sequence {(u n , v n )} has a strongly convergent subsequence in B.

      Proof. By the fact c α 0 λ 1 - N p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq27_HTML.gif, we have
      c - I λ ( u , v ) α 0 λ 1 - N p - I λ ( u , v ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equam_HTML.gif

      This, together with I λ (u, v) ≥ 0 and Lemma 3.5, gives the desired conclusion. □

      Next, we consider λ = 1. From the following standard argument, we get that I λ possesses the mountain-pass structure.

      Lemma 3.7. Under the assumptions of Theorem 2, there exist α λ , ρ λ > 0 such that
      I λ ( u , v ) > 0 if 0 < | | ( u , v ) | | λ < ρ λ and  I λ ( u , v ) α λ if | | ( u , v ) | | λ = ρ λ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equan_HTML.gif
      Proof. By (3.5), we get that for any δ > 0, there is C δ > 0 such that
      N G ( u , v ) δ ( | | u | | p p + | | v | | p p ) + C δ ( | | u | | p * p * + | | v | | p * p * ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equao_HTML.gif
      Thus
      I λ ( u , v ) = 1 p | | ( u , v ) | | λ p - λ N G ( u , v ) 1 p | | ( u , v ) | | λ p - λ δ ( | | u | | p p + | | v | | p p ) - λ C δ ( | | u | | p * p * + | | v | | p * p * ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equap_HTML.gif
      Note that | | u | | p p + | | v | | p p C 1 | | ( u , v ) | | λ p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq28_HTML.gif. If δ ≤ (2pλC1)-1, then
      I λ ( u , v ) 1 2 p | | ( u , v ) | | λ p - λ C δ ( | | u | | p * p * + | | v | | p * p * ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equaq_HTML.gif

      The fact p* > p implies the desired conclusion. □

      Lemma 3.8. Under the assumptions of Lemma 3.7, for any finite dimensional subspace

      FB, we have
      I λ ( u , v ) - as  ( u , v ) F , | | ( u , v ) | | λ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equar_HTML.gif
      Proof. By the assumption (H3), it follows that
      I λ ( u , v ) 1 p | | ( u , v ) | | λ p - λ a 0 ( | u | α α + | v | β β ) for all  ( u , v ) B . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equas_HTML.gif

      Since all norms in a finite-dimensional space are equivalent and α, β > p, we prove the result of this Lemma. □

      By Lemma 3.6, for λ larger enough and c λ small sufficiently, I λ satisfies (PS) condition.

      Thus, we will find special finite-dimensional subspaces by which we establish sufficiently small minimax levels.

      Define the functional
      Φ λ ( u , v ) = 1 p N ( | u | p + λ V ( x ) | u | p + | v | p + λ V ( x ) | v | p ) - λ a 0 N ( | u | α + | v | β ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equat_HTML.gif

      It is apparent that Φ λ C1(B) and I λ (u, v) ≤ Φ λ (u, v) for all (u, v) ∈ B.

      Observe that
      inf N | ϕ | p : ϕ C 0 ( N , ) , | ϕ | L α ( N ) = 1 = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equau_HTML.gif
      and
      inf N | ψ | p : ψ C 0 ( N , ) , | ψ | L β ( N ) = 1 = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equav_HTML.gif

      For any δ> 0, there are φ δ , ψ δ C 0 ( N , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq29_HTML.gif with | ϕ δ | L α ( N ) = | ψ δ | L β ( N ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq30_HTML.gif and suppφδ, supp ψ δ B r δ ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq31_HTML.gif such that | ϕ δ | p p , | ψ δ | p p < δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq32_HTML.gif.

      Let w λ ( x ) = ( ϕ δ ( λ p x ) , ψ δ ( λ p x ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq33_HTML.gif, then supp w λ B λ - 1 p r δ ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq34_HTML.gif. For t ≥ 0, we get
      Φ λ ( t w λ ) = t p p w λ λ p - a 0 λ t α N | ϕ δ ( λ p x ) | α - a 0 λ t β N | ψ δ ( λ p x ) | β = λ 1 - N p J λ ( t ϕ δ , t ψ δ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equaw_HTML.gif
      where
      J λ ( u , v ) = 1 p N ( | u | p + | v | p + V ( λ - 1 p x ) ( | u | p + | v | p ) ) - a 0 N ( | u | α + | v | β ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equax_HTML.gif
      We easily prove that
      max t 0 J λ ( t ϕ δ , t ψ δ ) α p p α ( α a 0 ) p α p { N ( | ϕ δ | p + V ( λ 1 p x ) | ϕ δ | p } α α p + β p p β ( β a 0 ) p β p { N ( | ψ δ | p + V ( λ 1 p x ) | ψ δ | p } β β p . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equay_HTML.gif
      Together with V (0) = 0 and | ϕ δ | p p , | ψ δ | p p < δ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq35_HTML.gif, this implies that there is Λδ> 0 such that for all λ ≥ Λδ, we have
      max t 0 I λ ( t ϕ δ , t ψ δ ) α - p p α ( α a 0 ) p α - p ( 2 δ ) α α - p + β - p p β ( β a 0 ) p β - p ( 2 δ ) β β - p λ 1 - N p . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equ10_HTML.gif
      (3.6)

      It follows from (3.6) that

      Lemma 3.9. Under the assumptions of Lemma 3.7, for any ⊂ > 0, there is Λσ> 0 such that λ ≥ Λσ, there exists w ̄ λ B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq36_HTML.gif with w ̄ λ λ > ρ λ , I λ ( w ̄ λ ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq37_HTML.gif and
      max t 0 I λ ( t w ̄ λ ) σ λ 1 - N p , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equaz_HTML.gif

      where ρ λ is defined in Lemma 3.7.

      Proof. This proof is similar to the one of Lemma 4.3 in [10], it can be easily proved. □

      4 Proof of the main result

      In the following, we will give the proof of Theorem 2.

      Proof. From Lemma 3.9, for any σ > 0 with 0 < σ < α0, there is Λ σ > 0 such that for λ ≥ Λ σ , we obtain
      c λ = inf γ Γ λ max t [ 0 , 1 ] I λ ( γ ( t ) ) σ λ 1 - N p , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equba_HTML.gif

      where Γ λ = { γ C ( [ 0 , 1 ] , B ) : γ ( 0 ) = 0 , γ ( 1 ) = w ̄ λ } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq38_HTML.gif

      Furthermore, Lemma 3.6 implies that I λ satisfies (PS) condition. Hence, by the mountain-pass theorem, there is (u λ , v λ ) ∈ B satisfying I λ (u λ , v λ ) = c λ and I λ ( u λ , v λ ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq39_HTML.gif This shows (u λ , v λ ) is a weak solution of (2.1). Similar to the argument in [10], we also get that (u λ , v λ ) is a positive least energy solution.

      Finally, we prove (u λ , v λ ) satisfies the estimate (2.2). Observe that I λ ( u λ , v λ ) σ λ 1 - N p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq40_HTML.gif and I λ ( u λ , v λ ) = 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq41_HTML.gif we have
      I λ ( u λ , v λ ) = I λ ( u λ , v λ ) - 1 θ I λ ' ( u λ , v λ ) ( u λ , v λ ) = 1 p - 1 θ ( u λ , v λ ) λ p + 1 θ - 1 p * λ N K ( x ) ( | u λ | p * + | v λ | p * ) + λ N 1 θ ( u λ H s ( u λ , v λ ) + v λ H t ( u λ , v λ ) ) - H ( u λ , v λ ) 1 p - 1 θ ( u λ , v λ ) λ p . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_Equbb_HTML.gif

      This shows that (u λ , v λ ) satisfies the estimate (2.2). The proof is complete. □

      Declarations

      Acknowledgements

      The authors would like to appreciate the referees for their precious comments and suggestions about the original manuscript. This research is supported by the National Natural Science Foundation of China (10771212) and the Fundamental Research Funds for the Central Universities (2010LKSX09).

      Authors’ Affiliations

      (1)
      Department of Mathematics, China University of Mining and Technology

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      Copyright

      © Zhang and Liu; licensee Springer. 2012

      This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://​creativecommons.​org/​licenses/​by/​2.​0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.