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# 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

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

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

(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

(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

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

(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 .

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 denote the collection of smooth functions with compact support and D1,p(ℝN) be the completion of under

We introduce the space

equipped with the norm

and the space

under

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 for any (u, v) ∈ B. Let λ = ε-p in the system (1.1), then (1.1) is changed into

(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

(2.2)

The energy functional associated with (2.1) is defined by

where .

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

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

Together with Iλ(un, vn) → c and 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 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 such that for any ε > 0, there is rε > 0 with

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 , . It is obvious that

(3.1)

Lemma 3.3. One has

and

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

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

(3.2)

By Hölder inequality and Lemma 3.2, it follows that

and

Similarly, we get

and

Thus

From the similar argument, we also get

Lemma 3.4. One has along a subsequence

and

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

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

and

In connection with the fact Iλ (un, vn) → c and , we obtain

In the following, we will verify the fact .

For any (φ, ψ) ∈ B, it follows that

Standard argument shows that

and

uniformly in ‖φ, ψ)‖B 1.

By Lemma 3.3, we have

and

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

Let , , then , . From (3.1), we get (un, vn) → (u, v) in B if and only if in B.

Observe that

where .

Thus by Lemma 3.4, we get

(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 , in , we get

(3.4)

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

(3.5)

Let S be the best Sobolev constant of the immersion

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 .

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

and

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

This, together with and (3.3), gives

Set , then

This proof is completed. □

Since 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 , the sequence {(un, vn)} has a strongly convergent subsequence in B.

Proof. By the fact , we have

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

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

Thus

Note that . If δ ≤ (2pλC1)-1, then

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

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

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

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

Observe that

and

For any δ> 0, there are φδ, with and suppφδ, such that .

Let , then . For t ≥ 0, we get

where

We easily prove that

Together with V (0) = 0 and , this implies that there is Λδ > 0 such that for all λ ≥ Λδ, we have

(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 with and

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

where

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 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 and we have

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.

### 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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