Open Access

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

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 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 | , https://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) V C( N ), V (0) = inf V (x) = 0 and there exists b > 0 such that the set ν b := {x N : V (x) < b} has finite Lebesgue measure;

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

(H1) H C1(2) 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 ) ; https://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 . https://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 . https://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) V C( N ), min V = 0 and there is b > 0 such that the set ν b := {x N : V (x) < b} has finite Lebesgue measure;

(A2) K(x) C( N ), 0 < inf K ≤ sup K <

(B1) h C( N × ) 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 https://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 N . Moreover, Liu and Zheng [13] investigated (1.2) in 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 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 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 ) https://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( N ) be the completion of C 0 ( N ) https://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 . https://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 < } https://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 https://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 https://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 . https://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( N , V) continuously embeds in W1,p( N ).

Set B = E λ × E λ and | | ( u , v ) | | λ = | | u | | λ p + | | v | | λ p https://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 | . https://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 . https://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 ) , https://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 ) https://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, ) 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 ) https://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 . https://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 https://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 ) https://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 N .

Lemma 3.2. Let d [p, p*). There exists a subsequence { ( u n j , v n j ) } https://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 https://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 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 η C(+) 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 ) https://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 ) https://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 . https://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 https://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 https://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 https://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 https://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 https://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 https://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 . https://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 . https://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 . https://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 ) https://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 ) . https://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 ) . https://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 https://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 . https://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 ) https://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 ) . https://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 https://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 ) . https://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 https://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 https://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 https://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 https://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 . https://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 https://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 https://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 ) https://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 ) https://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 ) https://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 * ) , https://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 https://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 ) . https://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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-53/MediaObjects/13661_2011_Article_142_IEq19_HTML.gif, v n 1 0 https://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 ) https://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 ) . https://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 * ) . https://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 ) . https://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 https://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 https://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 . https://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 ) . https://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 https://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 ) . https://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 https://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 ) . https://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 ) https://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 https://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 https://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 ) . https://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 . https://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 * ) . https://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 * ) . https://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 https://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 * ) . https://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

F B, we have
I λ ( u , v ) - as  ( u , v ) F , | | ( u , v ) | | λ . https://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 . https://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 ) . https://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 https://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 . https://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 , ) https://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 https://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 ) https://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 https://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 ) ) https://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 ) https://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 ) , https://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 ) . https://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 . https://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 https://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 . https://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 https://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 https://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 , https://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 , https://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 } . https://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 . https://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 https://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 . https://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 . https://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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© 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.