Open Access Research

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

Huixing Zhang* and Wenbin Liu

Author Affiliations

Department of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu 221116, People's Republic of China

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Boundary Value Problems 2012, 2012:53  doi:10.1186/1687-2770-2012-53


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/53


Received:29 September 2011
Accepted:4 May 2012
Published:4 May 2012

© 2012 Zhang and Liu; licensee Springer.

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.

Abstract

We consider a p-Laplacian system with critical nonlinearity in ℝN. 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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M1">View MathML</a>

(1.1)

where Δpu = 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(ℝN), V (0) = inf V (x) = 0 and there exists b > 0 such that the set νb := {x ∈ ℝN : V (x) < b} has finite Lebesgue measure;

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

(H1) H C1(ℝ2) and Hs, Ht = o(|s|p-1 + |t|p-1) as |s| + |t| → 0;

(H2) there exist c > 0 and p < q < p* such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M2">View MathML</a>

(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) ≤ sHs + tHt.

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 εσ > 0 such that if ε < εσ, the problem (1.1) has at least one positive solution (uε, vε) which satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M3">View MathML</a>

The scalar form of the problem (1.1) is as follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M4">View MathML</a>

(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(ℝN), min V = 0 and there is b > 0 such that the set νb := {x ∈ ℝN : V (x) < b} has finite Lebesgue measure;

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

(B1) h C(ℝN × 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M5">View MathML</a>.

That article obtained the existence of at least one positive solution uε 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 p2 N 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 ⊂ ℝN. Moreover, Liu and Zheng [13] investigated (1.2) in ℝN 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 ℝN.

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 ℝN 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M6">View MathML</a> denote the collection of smooth functions with compact support and D1,p(ℝN) be the completion of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M7">View MathML</a> under

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M8">View MathML</a>

We introduce the space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M9">View MathML</a>

equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M10">View MathML</a>

and the space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M11">View MathML</a>

under

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M12">View MathML</a>

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

Set B = Eλ × Eλ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M13">View MathML</a> for any (u, v) ∈ B. Let λ = ε-p in the system (1.1), then (1.1) is changed into

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M14">View MathML</a>

(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 Λσ > 0 such that if λ ≥ Λσ , the system (2.1) has at least one least energy solution (uλ, vλ) satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M15">View MathML</a>

(2.2)

The energy functional associated with (2.1) is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M16">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M17">View MathML</a>.

From the assumptions of Theorem 2, standard arguments [14] show that Iλ 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 {(un, vn)} ⊂ B is a (PS)c sequence for Iλ, then we get that c ≥ 0 and {(un, vn)} is bounded in the space B.

Proof. One has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M18">View MathML</a>

By the assumptions (K0) and (H3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M19">View MathML</a>

Together with Iλ(un, vn) → c and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M20">View MathML</a> 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 (un, vn) ⇀ (u, v) in B. Furthermore, passing to a subsequence, we have un u and vn v in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M21">View MathML</a> for any d ∈ [p, p*) and un u, vn v a.e. in ℝN.

Lemma 3.2. Let d ∈ [p, p*). There exists a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M22">View MathML</a> such that for any ε > 0, there is rε > 0 with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M23">View MathML</a>

for any r ≥ rε , where Br := {x ∈ ℝN : |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(ℝ+) be a smooth function satisfying 0 ≤ η(t) 1, η(t) = 1 if t ≤ 1 and η(t) = 0 if t ≥ 2. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M25">View MathML</a>. It is obvious that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M26">View MathML</a>

(3.1)

Lemma 3.3. One has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M27">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M28">View MathML</a>

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

Proof. From the assumptions (H1)-(H2) and Lemma 3.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M29">View MathML</a>

(3.2)

By Hölder inequality and Lemma 3.2, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M30">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M31">View MathML</a>

Similarly, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M32">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M33">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M34">View MathML</a>

From the similar argument, we also get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M35">View MathML</a>

Lemma 3.4. One has along a subsequence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M36">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M37">View MathML</a>

Proof. From the Lemma 2.1 of [15] and the argument of [16], we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M38">View MathML</a>

By (3.1) and the similar idea of proving the Brézis-Lieb Lemma [17], it is easy to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M39">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M40">View MathML</a>

In connection with the fact Iλ (un, vn) → c and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M41">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M42">View MathML</a>

In the following, we will verify the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M43">View MathML</a>.

For any (φ, ψ) ∈ B, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M44">View MathML</a>

Standard argument shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M45">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M46">View MathML</a>

uniformly in ‖φ, ψ)‖B 1.

By Lemma 3.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M47">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M48">View MathML</a>

uniformly in ‖(φ, ψ)‖B 1. From the facts above mentioned, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M49">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M51">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M53">View MathML</a>. From (3.1), we get (un, vn) → (u, v) in B if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M54">View MathML</a> in B.

Observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M55">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M56">View MathML</a>.

Thus by Lemma 3.4, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M57">View MathML</a>

(3.3)

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

Let Vb(x):= max{V (x), b}, where b is the positive constant in the assumption (V0). Since the set νb has finite measure and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M59">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M60">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M61">View MathML</a>

(3.4)

From (K0), (H1)-(H3) and Young inequality, there is Cb > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M62">View MathML</a>

(3.5)

Let S be the best Sobolev constant of the immersion

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M63">View MathML</a>

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 {(un, vn)} ⊂ B for Iλ with (un, vn) ⇀ (u, v), either (un, vn) (u, v) or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M64">View MathML</a>.

Proof. Assume that (un, vn) ↛ (u, v), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M65">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M66">View MathML</a>

By the Sobolev inequality, (3.4) and (3.5), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M67">View MathML</a>

This, together with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M68">View MathML</a> and (3.3), gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M69">View MathML</a>

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M70">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M71">View MathML</a>

This proof is completed. □

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M72">View MathML</a> is not compact, Iλ does not satisfy the (PS)c condition for all c > 0. But Lemma 3.5 shows that Iλ 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 {(un, vn)} ⊂ B for Iλ satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M73">View MathML</a>, the sequence {(un, vn)} has a strongly convergent subsequence in B.

Proof. By the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M74">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M75">View MathML</a>

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

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

Lemma 3.7. Under the assumptions of Theorem 2, there exist αλ, ρλ > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M76">View MathML</a>

Proof. By (3.5), we get that for any δ > 0, there is Cδ > 0 such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M77">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M78">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M79">View MathML</a>. If δ ≤ (2pλC1)-1, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M80">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M81">View MathML</a>

Proof. By the assumption (H3), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M82">View MathML</a>

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 cλ small sufficiently, Iλ satisfies (PS)condition.

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

Define the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M83">View MathML</a>

It is apparent that Φλ C1(B) and Iλ(u, v) ≤ Φλ (u, v) for all (u, v) ∈ B.

Observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M84">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M85">View MathML</a>

For any δ> 0, there are φδ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M86">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M87">View MathML</a> and suppφδ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M88">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M89">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M90">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M91">View MathML</a>. For t ≥ 0, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M92">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M93">View MathML</a>

We easily prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M94">View MathML</a>

Together with V (0) = 0 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M95">View MathML</a>, this implies that there is Λδ > 0 such that for all λ ≥ Λδ, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M96">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M97">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M98">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M99">View MathML</a>

where ρλ 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 Λσ > 0 such that for λ ≥ Λσ, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M100">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M101">View MathML</a>

Furthermore, Lemma 3.6 implies that Iλ satisfies (PS)condition. Hence, by the mountain-pass theorem, there is (uλ, vλ) ∈ B satisfying Iλ (uλ, vλ) = cλ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M102">View MathML</a> This shows (uλ, vλ) is a weak solution of (2.1). Similar to the argument in [10], we also get that (uλ, vλ) is a positive least energy solution.

Finally, we prove (uλ, vλ) satisfies the estimate (2.2). Observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M103">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M104">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/53/mathml/M105">View MathML</a>

This shows that (uλ, vλ) satisfies the estimate (2.2). The proof is complete. □

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

The authors contributed equally in this article. They read and approved the final manuscript.

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).

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