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

Huixing Zhang; Wenbin Liu

doi:10.1186/1687-2770-2012-53

Research

Open Access

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

Huixing Zhang¹Email author and
Wenbin Liu¹

Boundary Value Problems20122012:53

https://doi.org/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 systemcritical nonlinearityvariational methods.

1 Introduction

This article is concerned with the existence of solutions to the following nonlinear perturbed p-Laplacian system

{\begin{array}{l} - ε^{p} Δ_{p} u + V (x) | u |^{p - 2} u = K (x) | u |^{p^{*} - 2} u + H_{u} (u, v), x \in ℝ^{N}, \\ - ε^{p} Δ_{p} v + V (x) | v |^{p - 2} v = K (x) | v |^{p^{*} - 2} v + H_{v} (u, v), x \in ℝ^{N}, \\ u (x), v (x) > 0, \\ u (x), v (x) \to 0 as | x | \to \infty, \end{array}

(1.1)

where Δ_pu = div(|∇u|^p-2∇u) 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(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) H ∈ C¹(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) | \leq c (1 + | s |^{q - 1} + | t |^{q - 1});

(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

\frac{θ - p}{p θ} \int_{ℝ^{N}} (ε^{p} | \nabla u_{ε} |^{p} + ε^{p} | \nabla v_{ε} |^{p} + V (x) | u_{ε} |^{p} + V (x) | v_{ε} |^{p}) \leq σ ε^{N} .

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 \in ℝ^{N} .

(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(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) h ∈ C(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 $H (x, u) = \int_{0}^{u} h (x, s) d s$ .

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 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 ⊂ 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}^{\infty} (ℝ^{N})

denote the collection of smooth functions with compact support and D^1,p(ℝ ^N ) be the completion of

C_{0}^{\infty} (ℝ^{N})

under

| | u | |^{p} = \int_{ℝ^{N}} | \nabla u |^{p} d x .

We introduce the space

E (ℝ^{N}, V) = {u \in W^{1, p} (ℝ^{N}) : \int_{ℝ^{N}} V (x) | u |^{p} < \infty}

equipped with the norm

| | u | |_{E} = {(\int_{ℝ^{N}} (| \nabla u |^{p} + V (x) | u |^{p}))}^{\frac{1}{p}}

and the space

E_{λ} (ℝ^{N}, V) = \{u \in W^{1, p} (ℝ^{N}) : \int_{ℝ^{N}} λ V (x) | u |^{p} < \infty, λ > 0\}

under

| | u | |_{λ} = {(\int_{ℝ^{N}} | \nabla u |^{p} + λ V (x) | u |^{p}))}^{\frac{1}{p}} .

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 W^1,p(ℝ ^N ).

Set B = E_λ × E_λ and

| | (u, v) | |_{λ} = | | u | |_{λ}^{p} + | | v | |_{λ}^{p}

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

{\begin{array}{l} - Δ_{p} u + λ V (x) | u |^{p - 2} u = λ K (x) | u |^{p^{*} - 2} u + λ H_{u} (u, v), \in ℝ^{N}, \\ - Δ_{p} v + λ V (x) | v |^{p - 2} v = λ K (x) | v |^{p^{*} - 2} v + λ H_{v} (u, v), x \in ℝ^{N}, \\ u (x), v (x) > 0, \\ u (x), v (x) \to 0, as | x | \to \infty . \end{array}

(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

\frac{θ - p}{p θ} \int_{ℝ^{N}} (| \nabla u_{λ} |^{p} + | \nabla v_{λ} |^{p} + λ V (x) (| u_{λ} |^{p} + | v_{λ} |^{p})) \leq σ λ^{1 - \frac{N}{p}} .

(2.2)

The energy functional associated with (2.1) is defined by

\begin{array}{l} I_{λ} (u, v) & = \frac{1}{p} \int_{ℝ^{N}} (| \nabla u |^{p} + λ V (x) | u |^{p} + | \nabla v |^{p} + λ V (x) | v |^{p}) \\ - \frac{λ}{p^{*}} \int_{ℝ^{N}} K (x) (| u |^{p^{*}} + | v |^{p^{*}}) - λ \int_{ℝ^{N}} H (u, v) \\ = \frac{1}{p} | | (u, v) | |_{λ}^{p} - λ \int_{ℝ^{N}} G (u, v), \end{array}

where $G (u, v) = \frac{1}{p^{*}} K (x) (| u |^{p *} + | v |^{p *}) + H (u, v)$ .

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

\begin{array}{l} I_{λ} (u_{n}, v_{n}) - \frac{1}{θ} I_{λ}^{'} (u_{n}, v_{n}) (u_{n}, v_{n}) \\ = \frac{1}{p} | | (u_{n}, v_{n}) | |_{λ}^{p} - \frac{λ}{p^{*}} \int_{ℝ^{N}} K (x) (| u_{n} |^{p^{*}} + | v_{n} |^{p^{*}}) - λ \int_{ℝ^{N}} H (u_{n}, v_{n}) \\ - \frac{1}{θ} [| | (u_{n}, v_{n}) | |_{λ}^{p} - λ \int_{ℝ^{N}} K (x) (| u_{n} |^{p^{*}} + | v_{n} |^{p^{*}}) - λ \int_{ℝ^{N}} (u_{n} H_{s} (u_{n}, v_{n}) + v_{n} H_{t} (u_{n}, v_{n}))] \\ = (\frac{1}{p} - \frac{1}{θ}) | | (u_{n}, v_{n}) | |_{λ}^{p} + (\frac{1}{θ} - \frac{1}{p^{*}}) λ \int_{ℝ^{N}} K (x) (| u_{n} |^{p^{*}} + | v_{n} |^{p^{*}}) \\ + λ \int_{ℝ^{N}} (\frac{1}{θ} (u_{n} H_{s} (u_{n}, v_{n}) + v_{n} H_{t} (u_{n}, v_{n})) - H (u_{n}, v_{n})) \end{array}

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

I_{λ} (u_{n}, v_{n}) - \frac{1}{θ} I_{λ}^{'} (u_{n}, v_{n}) (u_{n}, v_{n}) \geq (\frac{1}{p} - \frac{1}{θ}) | | (u_{n}, v_{n}) | |_{λ}^{p} .

Together with I_λ (u_n , v_n ) → c and $I_{λ}^{'} (u_{n}, v_{n}) \to 0$ 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 $L_{l o c}^{d} (ℝ^{N})$ 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}})}

such that for any ε > 0, there is r_ε > 0 with

lim_{i \to \infty} sup \int_{B_{i} \ B_{r}} (| u_{n_{i}} |^{d} + | v_{n_{i}} |^{d}) \leq ε

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 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

ũ_{j} (x) = η (2 | x | / j) u (x)

,

ṽ_{j} (x) = η (2 | x | / j) v (x)

. It is obvious that

| | u - ũ_{j} | |_{λ} \to 0 and | | v - ṽ_{j} | |_{λ} \to 0 as j \to \infty .

(3.1)

Lemma 3.3. One has

lim_{j \to \infty} \int_{ℝ^{N}} (H_{s} (u_{n_{j}}, v_{n_{j}}) - H_{s} (u_{n_{j}} - ũ_{j}, v_{n_{j}} - ṽ_{j}) - H_{s} (ũ_{j}, ṽ_{j})) φ = 0

and

lim_{j \to \infty} \int_{ℝ^{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}, ṽ_{j})) ψ = 0

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

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

\begin{array}{l} \lim_{j \to \infty} \sup \int_{ℝ^{N}} (H_{s} (u_{n_{j}}, v_{n_{j}}) - H_{s} (u_{n_{j}} - {\tilde{u}}_{j}, v_{n_{j}} - {\tilde{v}}_{j}) - H_{s} ({\tilde{u}}_{j}, {\tilde{v}}_{j})) φ \\ = \lim_{j \to \infty} \sup \int_{B_{j}} (H_{s} (u_{n_{j}}, v_{n_{j}}) - H_{s} (u_{n_{j}} - {\tilde{u}}_{j}, v_{n_{j}} - {\tilde{v}}_{j}) - H_{s} ({\tilde{u}}_{j}, {\tilde{v}}_{j})) φ \\ = \lim_{j \to \infty} \sup \int_{B_{j} \ B_{r}} (H_{s} (u_{n_{j}}, v_{n_{j}}) - H_{s} (u_{n_{j}} - {\tilde{u}}_{j}, v_{n_{j}} - {\tilde{v}}_{j}) - H_{s} ({\tilde{u}}_{j}, {\tilde{v}}_{j})) φ \\ \leq c \lim_{j \to \infty} \sup \int_{B_{j} \ B_{r}} (| u_{n_{j}} |^{p - 1} + | v_{n_{j}} |^{p - 1} + | u_{n_{j}} |^{q - 1} + | v_{n_{j}} |^{q - 1} + | {\tilde{u}}_{j} |^{p - 1} + | {\tilde{v}}_{j} |^{p - 1} \\ + | {\tilde{u}}_{j} |^{q - 1} + | {\tilde{v}}_{j} |^{q - 1} + | u_{n_{j}} - {\tilde{u}}_{j} |^{p - 1} + | v_{n_{j}} - {\tilde{v}}_{j} |^{p - 1} + | u_{n_{j}} - {\tilde{u}}_{j} |^{q - 1} + | v_{n_{j}} - {\tilde{v}}_{j} |^{q - 1}) φ \\ \leq c_{1} \lim_{j \to \infty} \sup \int_{B_{j} \ B_{r}} (| u_{n_{j}} |^{p - 1} + | v_{n_{j}} |^{p - 1} + | {\tilde{u}}_{j} |^{p - 1} + | {\tilde{v}}_{j} |^{p - 1}) φ \\ + c_{2} \lim_{j \to \infty} \sup \int_{B_{j} \ B_{r}} (| u_{n_{j}} |^{q - 1} + | v_{n_{j}} |^{q - 1} + | {\tilde{u}}_{j} |^{q - 1} + | {\tilde{v}}_{j} |^{q - 1}) φ \end{array}

(3.2)

By Hölder inequality and Lemma 3.2, it follows that

\begin{array}{l} lim_{j \to \infty} sup \int_{B_{j} \ B_{r}} | u_{n_{j}} |^{p - 1} | φ | & \leq lim_{j \to \infty} sup {(\int_{B_{j} \ B_{r}} | u_{n_{j}} |^{p})}^{\frac{p - 1}{p}} {(\int_{B_{j} \ B_{r}} | φ |^{p})}^{\frac{1}{p}} \\ \leq lim_{j \to \infty} sup {(\int_{B_{j} \ B_{r}} | u_{n_{j}} |^{p})}^{\frac{p - 1}{p}} {(\int_{ℝ^{N}} | φ |^{p})}^{\frac{1}{p}} \\ \leq lim_{j \to \infty} sup {(\int_{B_{j} \ B_{r}} | u_{n_{j}} |^{p})}^{\frac{p - 1}{p}} \\ = 0 \end{array}

and

\begin{array}{l} lim_{j \to \infty} sup \int_{B_{j} \ B_{r}} | u_{n_{j}} |^{p - 1} | φ | & \leq lim_{j \to \infty} sup {(\int_{B_{j} \ B_{r}} | u_{n_{j}} |^{p})}^{\frac{q - 1}{p}} {(\int_{B_{j} \ B_{r}} | φ |^{q})}^{\frac{1}{q}} \\ \leq lim_{j \to \infty} sup {(\int_{B_{j} \ B_{r}} | u_{n_{j}} |^{q})}^{\frac{q - 1}{q}} {(\int_{ℝ^{N}} | φ |^{q})}^{\frac{1}{q}} \\ \leq lim_{j \to \infty} sup {(\int_{B_{j} \ B_{r}} | u_{n_{j}} |^{q})}^{\frac{q - 1}{q}} \\ = 0 \end{array}

Similarly, we get

lim_{j \to \infty} sup \int_{B_{j} \ B_{r}} (| v_{n_{j}} |^{p - 1} | + | ũ_{j} |^{p - 1} + | ṽ_{j} |^{p - 1}) φ = 0

and

lim_{j \to \infty} sup \int_{B_{j} \ B_{r}} (| v_{n_{j}} |^{q - 1} | + | ũ_{j} |^{q - 1} + | ṽ_{j} |^{q - 1}) φ = 0 .

Thus

lim_{j \to \infty} \int_{ℝ^{N}} (H_{s} (u_{n_{j}}, v_{n_{j}}) - H_{s} (u_{n_{j}} - ũ_{j}, v_{n_{j}} - ṽ_{j}) - H_{s} (ũ_{j}, ṽ_{j})) φ = 0 .

From the similar argument, we also get

lim_{j \to \infty} \int_{ℝ^{N}} (H_{t} (u_{n_{j}}, v_{n_{j}}) - H_{t} (u_{n_{j}} - ũ_{j}, v_{n_{j}} - ṽ_{j}) - H_{t} (ũ_{j}, {ṽ^{'}}_{j})) ψ = 0 .

□

Lemma 3.4. One has along a subsequence

I_{λ} (u_{n} - ũ_{n}, v_{n} - ṽ_{n}) \to c - I_{λ} (u, v)

and

I_{λ}^{'} (u_{n} - ũ_{n}, v_{n} - ṽ_{n}) \to 0 in B^{- 1} (the dual space of B) .

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

\begin{array}{l} I_{λ} (u_{n} - ũ_{n}, v_{n} - ṽ_{n}) \\ = \frac{1}{p} \int_{ℝ^{N}} (| \nabla u_{n} - \nabla ũ_{n} |^{p} + λ V (x) | u_{n} - ũ_{n} |^{p} + | \nabla v_{n} - \nabla ṽ_{n} |^{p} + λ V (x) | v_{n} - ṽ_{n} |^{p}) \\ - \frac{λ}{p^{*}} \int_{ℝ^{N}} K (x) (| u_{n} - ũ_{n} |^{p^{*}} + | v_{n} - ṽ_{n} |^{p^{*}}) - λ \int_{ℝ^{N}} H (u_{n} - ũ_{n}, v_{n} - ṽ_{n}) \\ = I_{λ} (u_{n}, v_{n}) - I_{λ} (ũ_{n}, ṽ_{n}) \\ + \frac{λ}{p^{*}} \int_{ℝ^{N}} K (x) ((| u_{n} |^{p^{*}} - | u_{n} - ũ_{n} |^{p^{*}} - | ũ_{n} |^{p^{*}}) + (| v_{n} |^{p^{*}} - | v_{n} - ṽ_{n} |^{p^{*}} - | ṽ_{n} |^{p^{*}})) \\ + λ \int_{ℝ^{N}} (H (u_{n}, v_{n}) - H (u_{n} - ũ_{n}, v_{n} - ṽ_{n}) - H (ũ_{n}, ṽ_{n})) + o (1) . \end{array}

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

lim_{n \to \infty} \int_{ℝ^{N}} K (x) ((| u_{n} |^{p^{*}} - | u_{n} - ũ_{n} |^{p^{*}} - | ũ_{n} |^{p^{*}}) + (| v_{n} |^{p^{*}} - | v_{n} - ṽ_{n} |^{p^{*}} - | ṽ_{n} |^{p^{*}})) = 0

and

lim_{n \to \infty} \int_{ℝ^{N}} (H (u_{n}, v_{n}) - H (u_{n} - ũ_{n}, v_{n} ṽ_{n}) - H (ũ_{n}, ṽ_{n})) = 0 .

In connection with the fact I_λ (u_n , v_n ) → c and

I_{λ} (ũ_{n}, ṽ_{n}) \to I_{λ} (u, v)

, we obtain

I_{λ} (u_{n} - ũ_{n}, v_{n} - ṽ_{n}) \to c - I_{λ} (u, v) .

In the following, we will verify the fact $I_{λ}^{'} (u_{n} - ũ_{n}, v_{n} - ṽ_{n}) \to 0$ .

For any (φ, ψ) ∈ B, it follows that

\begin{array}{l} I_{λ}^{'} (u_{n} - {\tilde{u}}_{n}, v_{n} - {\tilde{v}}_{n}) (φ, ψ) \\ = I_{λ}^{'} (u_{n}, v_{n}) (φ, ψ) - I_{λ}^{'} ({\tilde{u}}_{n}, {\tilde{v}}_{n}) (φ, ψ) \\ + λ \int_{ℝ^{N}} K (x) [(| u_{n} |^{p^{*} - 2} u_{n} - | u_{n} - {\tilde{u}}_{n} |^{p^{*} - 2} (u_{n} - {\tilde{u}}_{n}) - | {\tilde{u}}_{n} |^{p^{*} - 2} {\tilde{u}}_{n}) φ \\ + (| v_{n} |^{p^{*} - 2} v_{n} - | v_{n} - {\tilde{v}}_{n} |^{p^{*} - 2} (v_{n} - {\tilde{v}}_{n}) - | {\tilde{v}}_{n} |^{p^{*} - 2} {\tilde{v}}_{n}) ψ] \\ + λ \int_{ℝ^{N}} [(H_{s} (u_{n}, v_{n}) - H_{s} (u_{n} - {\tilde{u}}_{n}, v_{n} - {\tilde{v}}_{n}) - H_{s} ({\tilde{u}}_{n}, {\tilde{v}}_{n})) φ \\ + (H_{t} (u_{n}, v_{n}) - H_{t} (u_{n} - {\tilde{u}}_{n}, v_{n} - {\tilde{v}}_{n}) - H_{t} ({\tilde{u}}_{n}, {\tilde{v}}_{n})) ψ] + o (1) . \end{array}

Standard argument shows that

lim_{n \to \infty} \int_{ℝ^{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

and

lim_{n \to \infty} \int_{ℝ^{N}} K (x) (| v_{n} |^{p^{*} - 2} v_{n} - | v_{n} - ṽ_{n} |^{p^{*} - 2} (v_{n} - ṽ_{n}) - | ṽ_{n} |^{p^{*} - 2} ṽ_{n}) ψ = 0

uniformly in ‖φ, ψ)‖ _B ≤ 1.

By Lemma 3.3, we have

lim_{n \to \infty} \int_{ℝ^{N}} (H_{s} (u_{n}, v_{n}) - H_{s} (u_{n} - ũ_{n}, v_{n} - ṽ_{n}) - H_{s} (ũ_{n}, ṽ_{n})) φ = 0

and

lim_{n \to \infty} \int_{ℝ^{N}} (H_{t} (u_{n}, v_{n}) - H_{t} (u_{n} - ũ_{n}, v_{n} - ṽ_{n}) - H_{t} (ũ_{n}, ṽ_{n})) ψ = 0

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

I_{λ}^{'} (u_{n} - ũ_{n}, v_{n} - ṽ_{n}) \to 0 in B^{- 1} .

□

Let $u_{n}^{1} = u_{n} - ũ_{n}$ , $v_{n}^{1} = v_{n} - ṽ_{n}$ , then $u_{n} - u = u_{n}^{1} + (ũ_{n} - u)$ , $v_{n} - v = v_{n}^{1} + (ṽ_{n} - v)$ . From (3.1), we get (u_n , v_n ) → (u, v) in B if and only if $(u_{n}^{1}, v_{n}^{1}) \to (0, 0)$ in B.

Observe that

\begin{array}{l} I_{λ} (u_{n}^{1}, v_{n}^{1}) - \frac{1}{p} I_{λ}^{'} (u_{n}^{1}, v_{n}^{1}) (u_{n}^{1}, v_{n}^{1}) \\ = (\frac{1}{p} - \frac{1}{p^{*}}) λ \int_{ℝ^{N}} K (x) (| u_{n}^{1} |^{p^{*}} + | v_{n}^{1} |^{p^{*}}) \\ + λ \int_{ℝ^{N}} (\frac{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})) \\ \geq \frac{λ}{N} \int_{ℝ^{N}} K (x) (| u_{n}^{1} |^{p^{*}} + | v_{n}^{1} |^{p^{*}}) \\ \geq \frac{λ}{N} K_{min} \int_{ℝ^{N}} (| u_{n}^{1} |^{p^{*}} + | v_{n}^{1} |^{p^{*}}), \end{array}

where $K_{\min} = \inf_{x \in ℝ^{N}} K (x) > 0$ .

Thus by Lemma 3.4, we get

| | (u_{n}^{1}, v_{n}^{1}) | |_{p^{*}}^{p^{*}} \leq \frac{N (c - I_{λ} (u, v))}{λ K_{min}} + o (1) .

(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}

,

v_{n}^{1} \to 0

in

L_{loc}^{p} (ℝ^{N})

, we get

\int_{ℝ^{N}} V (x) (| u_{n}^{1} |^{p} + | v_{n}^{1} |^{p}) = \int_{ℝ^{N}} V_{b} (x) (| u_{n}^{1} |^{p} + | v_{n}^{1} |^{p}) + o (1) .

(3.4)

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

\begin{array}{l} \int_{ℝ^{N}} (K (x) (| u |^{p^{*}} + | v |^{p^{*}}) + u H_{s} (u, v) + v H_{t} (u, v)) \\ \leq & b (| | u | |_{p}^{p} + | | v | |_{p}^{p}) + C_{b} (| | u | |_{p^{*}}^{p^{*}} + | | v | |_{p^{*}}^{p^{*}}) . \end{array}

(3.5)

Let S be the best Sobolev constant of the immersion

S | | u | |_{p^{*}}^{p} \leq \int_{ℝ^{N}} | \nabla u |^{p} for all u \in W^{1, p} (ℝ^{N}) .

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 $c - I_{λ} (u, v) \geq α_{0} λ^{1 - \frac{N}{p}}$ .

Proof. Assume that (u_n , v_n ) ↛ (u, v), then

\lim \inf_{n \to \infty} | | (u_{n}^{1}, v_{n}^{1}) | |_{λ} > 0

and

c - J_{λ} (u, v) > 0 .

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

\begin{array}{l} S (| | u_{n}^{1} | |_{p^{*}}^{p} + | | v_{n}^{1} | |_{p^{*}}^{p}) \\ \leq \int_{ℝ^{N}} (| \nabla u_{n}^{1} |^{p} + | \nabla v_{n}^{1} |^{p}) \\ = \int_{ℝ^{N}} (| \nabla u_{n}^{1} |^{p} + λ V (x) | u_{n}^{1} |^{p} + | \nabla v_{n}^{1} |^{p} + λ V (x) | v_{n}^{1} |^{p}) - λ \int_{ℝ^{N}} V (x) (| u_{n}^{1} |^{p} + | v_{n}^{1} |^{p}) \\ = λ \int_{ℝ^{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 \int_{ℝ^{N}} V (x) (| u_{n}^{1} |^{p} + | v_{n}^{1} |^{p}) + o (1) \\ \leq 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) . \end{array}

This, together with

\lim \inf_{n \to \infty} (| | u_{n}^{1} | |_{p^{*}}^{p^{*}} + | | v_{n}^{1} | |_{p^{*}}^{p^{*}}) > 0

and (3.3), gives

\begin{array}{l} S & \leq λ C_{b} {(| | u_{n}^{1} | |_{p^{*}}^{p^{*}} + | | v_{n}^{1} | |_{p^{*}}^{p^{*}})}^{\frac{p^{*} - p}{p^{*}}} + o (1) \\ \leq λ C_{b} {(\frac{N (c - I_{λ} (u, v))}{λ K_{min}})}^{\frac{p}{N}} + o (1) \\ = λ^{1 - \frac{p}{N}} C_{b} {(\frac{N}{K_{min}})}^{\frac{p}{N}} {(c - I_{λ} (u, v))}^{\frac{p}{N}} + o (1) . \end{array}

Set

α_{0} = S^{\frac{N}{p}} C_{b}^{- \frac{N}{p}} N^{- 1} K_{min}

, then

α_{0} λ^{1 - \frac{N}{p}} \leq c - I_{λ} (u, v) + o (1) .

This proof is completed. □

Since $W^{1, p} (ℝ^{N}) ↪ L^{p^{*}} (ℝ^{N})$ 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 $c \leq α_{0} λ^{1 - \frac{N}{p}}$ , the sequence {(u_n , v_n )} has a strongly convergent subsequence in B.

Proof. By the fact

c \leq α_{0} λ^{1 - \frac{N}{p}}

, we have

c - I_{λ} (u, v) \leq α_{0} λ^{1 - \frac{N}{p}} - I_{λ} (u, v) .

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) \geq α_{λ} if | | (u, v) | |_{λ} = ρ_{λ} .

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

\int_{ℝ^{N}} G (u, v) \leq δ (| | u | |_{p}^{p} + | | v | |_{p}^{p}) + C_{δ} (| | u | |_{p^{*}}^{p^{*}} + | | v | |_{p^{*}}^{p^{*}}) .

Thus

\begin{array}{l} I_{λ} (u, v) & = \frac{1}{p} | | (u, v) | |_{λ}^{p} - λ \int_{ℝ^{N}} G (u, v) \\ \geq \frac{1}{p} | | (u, v) | |_{λ}^{p} - λ δ (| | u | |_{p}^{p} + | | v | |_{p}^{p}) - λ C_{δ} (| | u | |_{p^{*}}^{p^{*}} + | | v | |_{p^{*}}^{p^{*}}) . \end{array}

Note that

| | u | |_{p}^{p} + | | v | |_{p}^{p} \leq C_{1} | | (u, v) | |_{λ}^{p}

. If δ ≤ (2pλC₁)^-1, then

I_{λ} (u, v) \geq \frac{1}{2 p} | | (u, v) | |_{λ}^{p} - λ C_{δ} (| | u | |_{p^{*}}^{p^{*}} + | | v | |_{p^{*}}^{p^{*}}) .

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) \to - \infty as (u, v) \in F, | | (u, v) | |_{λ} \to \infty .

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

I_{λ} (u, v) \leq \frac{1}{p} | | (u, v) | |_{λ}^{p} - λ a_{0} (| u |_{α}^{α} + | v |_{β}^{β}) for all (u, v) \in B .

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) _cλ condition.

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

Define the functional

Φ_{λ} (u, v) = \frac{1}{p} \int_{ℝ^{N}} (| \nabla u |^{p} + λ V (x) | u |^{p} + | \nabla v |^{p} + λ V (x) | v |^{p}) - λ a_{0} \int_{ℝ^{N}} (| u |^{α} + | v |^{β}) .

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 \{\int_{ℝ^{N}} | \nabla ϕ |^{p} : ϕ \in C_{0}^{\infty} (ℝ^{N}, ℝ), | ϕ |_{L^{α} (ℝ^{N})} = 1\} = 0

and

inf \{\int_{ℝ^{N}} | \nabla ψ |^{p} : ψ \in C_{0}^{\infty} (ℝ^{N}, ℝ), | ψ |_{L^{β} (ℝ^{N})} = 1\} = 0 .

For any δ> 0, there are φ_δ , $ψ_{δ} \in C_{0}^{\infty} (ℝ^{N}, ℝ)$ with $| ϕ_{δ} |_{L^{α} (ℝ^{N})} = | ψ_{δ} |_{L^{β} (ℝ^{N})} = 1$ and suppφ_δ, $supp ψ_{δ} \subset B_{r_{δ}} (0)$ such that $| \nabla ϕ_{δ} |_{p}^{p}, | \nabla ψ_{δ} |_{p}^{p} < δ$ .

Let

w_{λ} (x) = (ϕ_{δ} (\sqrt[p]{λ} x), ψ_{δ} (\sqrt[p]{λ} x))

, then

supp w_{λ} \subset B_{λ^{- \frac{1}{p} r_{δ}}} (0)

. For t ≥ 0, we get

\begin{array}{l} Φ_{λ} (t w_{λ}) & = \frac{t^{p}}{p} {∥w_{λ}∥}_{λ}^{p} - a_{0} λ t^{α} \int_{ℝ^{N}} | ϕ_{δ} (\sqrt[p]{λ} x) |^{α} - a_{0} λ t^{β} \int_{ℝ^{N}} | ψ_{δ} (\sqrt[p]{λ} x) |^{β} \\ = λ^{1 - \frac{N}{p}} J_{λ} (t ϕ_{δ}, t ψ_{δ}), \end{array}

where

J_{λ} (u, v) = \frac{1}{p} \int_{ℝ^{N}} (| \nabla u |^{p} + | \nabla v |^{p} + V (λ^{- \frac{1}{p}} x) (| u |^{p} + | v |^{p})) - a_{0} \int_{ℝ^{N}} (| u |^{α} + | v |^{β}) .

We easily prove that

\begin{matrix} \max_{t \geq 0} J_{λ} (t ϕ_{δ}, t ψ_{δ}) \leq \frac{α - p}{p α {(α a_{0})}^{\frac{p}{α - p}}} {\int_{ℝ^{N}} (| \nabla ϕ_{δ} |^{p} + V (λ^{- \frac{1}{p}} x) | ϕ_{δ} |^{p}}^{\frac{α}{α - p}} \\ + \frac{β - p}{p β {(β a_{0})}^{\frac{p}{β - p}}} {\int_{ℝ^{N}} (| \nabla ψ_{δ} |^{p} + V (λ^{- \frac{1}{p}} x) | ψ_{δ} |^{p}}^{\frac{β}{β - p}} . \end{matrix}

Together with V (0) = 0 and

| \nabla ϕ_{δ} |_{p}^{p}, | \nabla ψ_{δ} |_{p}^{p} < δ

, this implies that there is Λ_δ> 0 such that for all λ ≥ Λ_δ, we have

max_{t \geq 0} I_{λ} (t ϕ_{δ}, t ψ_{δ}) \leq (\frac{α - p}{p α {(α a_{0})}^{\frac{p}{α - p}}} {(2 δ)}^{\frac{α}{α - p}} + \frac{β - p}{p β {(β a_{0})}^{\frac{p}{β - p}}} {(2 δ)}^{\frac{β}{β - p}}) λ^{1 - \frac{N}{p}} .

(3.6)

It follows from (3.6) that

Lemma 3.9. Under the assumptions of Lemma 3.7, for any ⊂ > 0, there is Λ_σ> 0 such that λ ≥ Λ_σ, there exists

{\bar{w}}_{λ} \in B

with

{∥{\bar{w}}_{λ}∥}_{λ} > ρ_{λ}, I_{λ} ({\bar{w}}_{λ}) \leq 0

and

max_{t \geq 0} I_{λ} (t {\bar{w}}_{λ}) \leq σ λ^{1 - \frac{N}{p}},

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 \in Γ_{λ}} max_{t \in [0, 1]} I_{λ} (γ (t)) \leq σ λ^{1 - \frac{N}{p}},

where $Γ_{λ} = {γ \in C ([0, 1], B) : γ (0) = 0, γ (1) = {\bar{w}}_{λ}} .$

Furthermore, Lemma 3.6 implies that I_λ satisfies (PS)_cλ condition. Hence, by the mountain-pass theorem, there is (u_λ , v_λ ) ∈ B satisfying I_λ (u_λ , v_λ ) = c_λ and $I_{λ}^{'} (u_{λ}, v_{λ}) = 0 .$ 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}) \leq σ λ^{1 - \frac{N}{p}}

and

I_{λ}^{'} (u_{λ}, v_{λ}) = 0 .

we have

\begin{array}{l} I_{λ} (u_{λ}, v_{λ}) & = I_{λ} (u_{λ}, v_{λ}) - \frac{1}{θ} I_{λ}^{'} (u_{λ}, v_{λ}) (u_{λ}, v_{λ}) \\ = (\frac{1}{p} - \frac{1}{θ}) {∥(u_{λ}, v_{λ})∥}_{λ}^{p} + (\frac{1}{θ} - \frac{1}{p^{*}}) λ \int_{ℝ^{N}} K (x) (| u_{λ} |^{p^{*}} + | v_{λ} |^{p^{*}}) \\ + λ \int_{ℝ^{N}} (\frac{1}{θ} (u_{λ} H_{s} (u_{λ}, v_{λ}) + v_{λ} H_{t} (u_{λ}, v_{λ})) - H (u_{λ}, v_{λ})) \\ \geq (\frac{1}{p} - \frac{1}{θ}) {∥(u_{λ}, v_{λ})∥}_{λ}^{p} . \end{array}

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, Xuzhou, People's Republic of China

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Boundary Value Problems

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