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

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.

1 Introduction

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

$$\{\begin{array}{l}-{\epsilon }^p{\Delta }_{p}u+V(x)|u{|}^{p-2}u=K(x)|u{|}^{p^{*}-2}u+H_u(u,v),x\in {ℝ}^N,\hfill \\ -{\epsilon }^p{\Delta }_{p}v+V(x)|v{|}^{p-2}v=K(x)|v{|}^{p^{*}-2}v+H_v(u,v),x\in {ℝ}^N,\hfill \\ u(x),v(x)>0,\hfill \\ u(x),v(x)\to 0\text{as}\left|x\right|\to \infty ,\hfill \end{array}$$

(1.1)

where Δ _p u = 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

$$\left|H_s\left(s,t\right)\right|,\left|H_t\left(s,t\right)\right|\le c\left(1+|s{|}^{q-1}+|t{|}^{q-1}\right);$$

(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

$$\frac{\theta -p}{p\theta }\underset{{ℝ}^N}{\int }\left({\epsilon }^p|\nabla u_{\epsilon }{|}^p+{\epsilon }^p|\nabla v_{\epsilon }{|}^p+V\left(x\right)|u_{\epsilon }{|}^p+V\left(x\right)|v_{\epsilon }{|}^p\right)\le \sigma {\epsilon }^N.$$

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

$$-{\epsilon }^p{\Delta }_{p}u+V\left(x\right)|u{|}^{p-2}u=K\left(x\right)|u{|}^{p^{*}-2}u+h\left(x,u\right),x\in {ℝ}^N.$$

(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 $H\left(x,u\right)={\int }_0^{u}h\left(x,s\right)ds$.

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 $C_0^{\infty }\left({ℝ}^N\right)$ denote the collection of smooth functions with compact support and D^1,p(ℝ ^N ) be the completion of $C_0^{\infty }\left({ℝ}^N\right)$ under

$$\left|\right|u|{|}^p=\underset{{ℝ}^N}{\int }|\nabla u{|}^{p}dx.$$

We introduce the space

$$E\left({ℝ}^N,V\right)=\left\{u\in W^{1,p}\left({ℝ}^N\right):\underset{{ℝ}^N}{\int }V\left(x\right)|u{|}^p<\infty \right\}$$

equipped with the norm

$$\left|\right|u|{|}_E={\left(\underset{{ℝ}^N}{\int }\left(|\nabla u{|}^p+V\left(x\right)|u{|}^p\right)\right)}^{\frac{1}{p}}$$

and the space

$$E_{\lambda }\left({ℝ}^N,V\right)=\left\{u\in W^{1,p}\left({ℝ}^N\right):\underset{{ℝ}^N}{\int }\lambda V\left(x\right)|u{|}^p<\infty ,\lambda >0\right\}$$

under

$$\left|\right|u|{|}_{\lambda }={\left({\displaystyle \underset{{ℝ}^N}{\int }|\nabla u{|}^p+\lambda V(x)|u{|}^p)}\right)}^{\frac{1}{p}}.$$

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 $\left|\right|\left(u,v\right)|{|}_{\lambda }=|\left|u\right|{|}_{\lambda }^p+\left|\right|v|{|}_{\lambda }^p$ for any (u, v) ∈ B. Let λ = ε^-pin the system (1.1), then (1.1) is changed into

$$\{\begin{array}{l}-{\Delta }_{p}u+\lambda V(x)|u{|}^{p-2}u=\lambda K(x)|u{|}^{p^{*}-2}u+\lambda H_u(u,v),\in {ℝ}^N,\hfill \\ -{\Delta }_{p}v+\lambda V(x)|v{|}^{p-2}v=\lambda K(x)|v{|}^{p^{*}-2}v+\lambda H_v(u,v),x\in {ℝ}^N,\hfill \\ u(x),v(x)>0,\hfill \\ u(x),v(x)\to 0,\text{as}\left|x\right|\to \infty .\hfill \end{array}$$

(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

$$\frac{\theta -p}{p\theta }\underset{{ℝ}^N}{\int }\left(|\nabla u_{\lambda }{|}^p+|\nabla v_{\lambda }{|}^p+\lambda V\left(x\right)\left(|u_{\lambda }{|}^p+|v_{\lambda }{|}^p\right)\right)\le \sigma {\lambda }^{1-\frac{N}{p}}.$$

(2.2)

The energy functional associated with (2.1) is defined by

$$\begin{array}{ll}\hfill I_{\lambda }\left(u,v\right)& =\frac{1}{p}\underset{{ℝ}^N}{\int }\left(|\nabla u{|}^p+\lambda V\left(x\right)|u{|}^p+|\nabla v{|}^p+\lambda V\left(x\right)|v{|}^p\right)\\ -\frac{\lambda }{p^{*}}\underset{{ℝ}^N}{\int }K\left(x\right)\left(|u{|}^{p^{*}}+|v{|}^{p^{*}}\right)-\lambda \underset{{ℝ}^N}{\int }H\left(u,v\right)\\ =\frac{1}{p}\left|\right|\left(u,v\right)|{|}_{\lambda }^p-\lambda \underset{{ℝ}^N}{\int }G\left(u,v\right),\end{array}$$

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

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

$$\begin{array}{l}{I}_{\lambda }\left(u_n,v_n\right)-\frac{1}{\theta }{I}_{\lambda }^{\text{'}}\left(u_n,v_n\right)\left(u_n,v_n\right)\\ =\frac{1}{p}\left|\right|\left(u_n,v_n\right)|{|}_{\lambda }^p-\frac{\lambda }{p^{*}}\underset{{ℝ}^N}{\int }K\left(x\right)\left(|u_n{|}^{p^{*}}+|v_n{|}^{p^{*}}\right)-\lambda \underset{{ℝ}^N}{\int }H\left(u_n,v_n\right)\\ -\frac{1}{\theta }\left[\left|\right|\left(u_n,v_n\right)|{|}_{\lambda }^p-\lambda \underset{{ℝ}^N}{\int }K\left(x\right)\left(|u_n{|}^{p^{*}}+|v_n{|}^{p^{*}}\right)-\lambda \underset{{ℝ}^N}{\int }\left(u_n{H}_s\left(u_n,v_n\right)+v_n{H}_t\left(u_n,v_n\right)\right)\right]\\ =\left(\frac{1}{p}-\frac{1}{\theta }\right)\left|\right|\left(u_n,v_n\right)|{|}_{\lambda }^p+\left(\frac{1}{\theta }-\frac{1}{p^{*}}\right)\lambda \underset{{ℝ}^N}{\int }K\left(x\right)\left(|u_n{|}^{p^{*}}+|v_n{|}^{p^{*}}\right)\\ +\lambda \underset{{ℝ}^N}{\int }\left(\frac{1}{\theta }\left(u_n{H}_s\left(u_n,v_n\right)+v_n{H}_t\left(u_n,v_n\right)\right)-H\left(u_n,v_n\right)\right)\end{array}$$

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

$$I_{\lambda }\left(u_n,v_n\right)-\frac{1}{\theta }{I}_{\lambda }^{\prime }\left(u_n,v_n\right)\left(u_n,v_n\right)\ge \left(\frac{1}{p}-\frac{1}{\theta }\right)\left|\right|\left(u_n,v_n\right)|{|}_{\lambda }^p.$$

Together with I _λ (u _n , v _n ) → c and $I_{\lambda }^{\prime }\left(u_n,v_n\right)\to 0$ 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 $L_{loc}^d\left({ℝ}^N\right)$ 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 $\left\{\left(u_{n_j},v_{n_j}\right)\right\}$ such that for any ε > 0, there is r _ε > 0 with

$$\underset{i\to \infty }{\text{lim}}\text{sup}\underset{B_i\backslash B_r}{\int }\left(|u_{n_i}{|}^d+|v_{n_i}{|}^d\right)\le \epsilon$$

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 ${ũ}_j\left(x\right)=\eta \left(2\left|x\right|/j\right)u\left(x\right)$, ${ṽ}_j\left(x\right)=\eta \left(2\left|x\right|/j\right)v\left(x\right)$. It is obvious that

$$\left|\right|u-{ũ}_j|{|}_{\lambda }\to 0\text{and}||v-{ṽ}_j|{|}_{\lambda }\to 0\text{as }j\to \infty .$$

(3.1)

Lemma 3.3. One has

$$\underset{j\to \infty }{\text{lim}}\underset{{ℝ}^N}{\int }\left(H_s\left(u_{n_j},v_{n_j}\right)-H_s\left(u_{n_j}-{ũ}_j,v_{n_j}-{ṽ}_j\right)-H_s\left({ũ}_j,{ṽ}_j\right)\right)\phi =0$$

and

$$\underset{j\to \infty }{\text{lim}}\underset{{ℝ}^N}{\int }\left(H_t\left(u_{n_j},v_{n_j}\right)-H_t\left(u_{n_j}-{ũ}_j,v_{n_j}-v_j\right)-H_t\left({ũ}_j,{ṽ}_j\right)\right)\psi =0$$

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

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

$$\begin{array}{l}\underset{j\to \infty }{\lim }\sup {\displaystyle \underset{{ℝ}^N}{\int }(H_s(u_{n_j},v_{n_j})-H_s(u_{n_j}-{\tilde{u}}_j,v_{n_j}-{\tilde{v}}_j)-H_s({\tilde{u}}_j,{\tilde{v}}_j))\phi }\\ =\underset{j\to \infty }{\lim }\sup {\displaystyle \underset{B_j}{\int }(H_s(u_{n_j},v_{n_j})-H_s(u_{n_j}-{\tilde{u}}_j,v_{n_j}-{\tilde{v}}_j)-H_s({\tilde{u}}_j,{\tilde{v}}_j))\phi }\\ =\underset{j\to \infty }{\lim }\sup {\displaystyle \underset{B_j\backslash B_r}{\int }(H_s(u_{n_j},v_{n_j})-H_s(u_{n_j}-{\tilde{u}}_j,v_{n_j}-{\tilde{v}}_j)-H_s(}{\tilde{u}}_j,{\tilde{v}}_j))\phi \\ \le c\underset{j\to \infty }{\lim }\sup {\displaystyle \underset{B_j\backslash B_r}{\int }(|u_{n_j}{|}^{p-1}+|v_{n_j}{|}^{p-1}+|u_{n_j}{|}^{q-1}+|v_{n_j}{|}^{q-1}+|{\tilde{u}}_j{|}^{p-1}+|{\tilde{v}}_j{|}^{p-1}}\\ +|{\tilde{u}}_j{|}^{q-1}+|{\tilde{v}}_j{|}^{q-1}+|u_{n_j}-{\tilde{u}}_j{|}^{p-1}+|v_{n_j}-{\tilde{v}}_j{|}^{p-1}+|u_{n_j}-{\tilde{u}}_j{|}^{q-1}+|v_{n_j}-{\tilde{v}}_j{|}^{q-1})\phi \\ \le c_1\underset{j\to \infty }{\lim }\sup {\displaystyle \underset{B_j\backslash B_r}{\int }(|u_{n_j}{|}^{p-1}+|v_{n_j}{|}^{p-1}+|{\tilde{u}}_j{|}^{p-1}+|{\tilde{v}}_j{|}^{p-1})\phi }\\ +c_2\underset{j\to \infty }{\lim }\sup {\displaystyle \underset{B_j\backslash B_r}{\int }(|u_{n_j}{|}^{q-1}+|v_{n_j}{|}^{q-1}+|{\tilde{u}}_j{|}^{q-1}+|{\tilde{v}}_j{|}^{q-1})\phi }\end{array}$$

(3.2)

By Hölder inequality and Lemma 3.2, it follows that

$$\begin{array}{ll}\hfill \underset{j\to \infty }{\text{lim}}\text{sup}\underset{B_j\backslash B_r}{\int }\left|u_{n_j}{|}^{p-1}\right|\phi |& \le \underset{j\to \infty }{\text{lim}}\text{sup}{\left(\underset{B_j\backslash B_r}{\int }|u_{n_j}{|}^p\right)}^{\frac{p-1}{p}}{\left(\underset{B_j\backslash B_r}{\int }|\phi {|}^p\right)}^{\frac{1}{p}}\\ \le \underset{j\to \infty }{\text{lim}}\text{sup}{\left(\underset{B_j\backslash B_r}{\int }|u_{n_j}{|}^p\right)}^{\frac{p-1}{p}}{\left(\underset{{ℝ}^N}{\int }|\phi {|}^p\right)}^{\frac{1}{p}}\\ \le \underset{j\to \infty }{\text{lim}}\text{sup}{\left(\underset{B_j\backslash B_r}{\int }|u_{n_j}{|}^p\right)}^{\frac{p-1}{p}}\\ =0\end{array}$$

and

$$\begin{array}{ll}\hfill \underset{j\to \infty }{\text{lim}}\text{sup}\underset{B_j\backslash B_r}{\int }\left|u_{n_j}{|}^{p-1}\right|\phi |& \le \underset{j\to \infty }{\text{lim}}\text{sup}{\left(\underset{B_j\backslash B_r}{\int }|u_{n_j}{|}^p\right)}^{\frac{q-1}{p}}{\left(\underset{B_j\backslash B_r}{\int }|\phi {|}^q\right)}^{\frac{1}{q}}\\ \le \underset{j\to \infty }{\text{lim}}\text{sup}{\left(\underset{B_j\backslash B_r}{\int }|u_{n_j}{|}^q\right)}^{\frac{q-1}{q}}{\left(\underset{{ℝ}^N}{\int }|\phi {|}^q\right)}^{\frac{1}{q}}\\ \le \underset{j\to \infty }{\text{lim}}\text{sup}{\left(\underset{B_j\backslash B_r}{\int }|u_{n_j}{|}^q\right)}^{\frac{q-1}{q}}\\ =0\end{array}$$

Similarly, we get

$$\underset{j\to \infty }{\text{lim}}\text{sup}\underset{B_j\backslash B_r}{\int }\left(\left|v_{n_j}{|}^{p-1}\right|+|{ũ}_j{|}^{p-1}+|{ṽ}_j{|}^{p-1}\right)\phi =0$$

and

$$\underset{j\to \infty }{\text{lim}}\text{sup}\underset{B_j\backslash B_r}{\int }\left(\left|v_{n_j}{|}^{q-1}\right|+|{ũ}_j{|}^{q-1}+|{ṽ}_j{|}^{q-1}\right)\phi =0.$$

Thus

$$\underset{j\to \infty }{\text{lim}}\underset{{ℝ}^N}{\int }\left(H_s\left(u_{n_j},v_{n_j}\right)-H_s\left(u_{n_j}-{ũ}_j,v_{n_j}-{ṽ}_j\right)-H_s\left({ũ}_j,{ṽ}_j\right)\right)\phi =0.$$

From the similar argument, we also get

$$\underset{j\to \infty }{\text{lim}}\underset{{ℝ}^N}{\int }\left(H_t\left(u_{n_j},v_{n_j}\right)-H_t\left(u_{n_j}-{ũ}_j,v_{n_j}-{ṽ}_j\right)-H_t\left({ũ}_j,{{ṽ}^{\prime }}_j\right)\right)\psi =0.$$

□

Lemma 3.4. One has along a subsequence

$$I_{\lambda }\left(u_n-{ũ}_n,v_n-{ṽ}_n\right)\to c-I_{\lambda }\left(u,v\right)$$

and

$$I_{\lambda }^{\prime }\left(u_n-{ũ}_n,v_n-{ṽ}_n\right)\to 0\text{in }{B}^{-1}\left(\text{the dual space of }B\right).$$

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

$$\begin{array}{l}{I}_{\lambda }\left(u_n-{ũ}_n,v_n-{ṽ}_n\right)\\ =\frac{1}{p}\underset{{ℝ}^N}{\int }\left(|\nabla u_n-\nabla {ũ}_n{|}^p+\lambda V\left(x\right)|u_n-{ũ}_n{|}^p+|\nabla v_n-\nabla {ṽ}_n{|}^p+\lambda V\left(x\right)|v_n-{ṽ}_n{|}^p\right)\\ -\frac{\lambda }{p^{*}}\underset{{ℝ}^N}{\int }K\left(x\right)\left(|u_n-{ũ}_n{|}^{p^{*}}+|v_n-{ṽ}_n{|}^{p^{*}}\right)-\lambda \underset{{ℝ}^N}{\int }H\left(u_n-{ũ}_n,v_n-{ṽ}_n\right)\\ =I_{\lambda }\left(u_n,v_n\right)-I_{\lambda }\left({ũ}_n,{ṽ}_n\right)\\ +\frac{\lambda }{p^{*}}\underset{{ℝ}^N}{\int }K\left(x\right)\left(\left(|u_n{|}^{p^{*}}-|u_n-{ũ}_n{|}^{p^{*}}-|{ũ}_n{|}^{p^{*}}\right)+\left(|v_n{|}^{p^{*}}-|v_n-{ṽ}_n{|}^{p^{*}}-|{ṽ}_n{|}^{p^{*}}\right)\right)\\ +\lambda \underset{{ℝ}^N}{\int }\left(H\left(u_n,v_n\right)-H\left(u_n-{ũ}_n,v_n-{ṽ}_n\right)-H\left({ũ}_n,{ṽ}_n\right)\right)+o\left(1\right).\end{array}$$

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

$$\underset{n\to \infty }{\text{lim}}\underset{{ℝ}^N}{\int }K\left(x\right)\left(\left(|u_n{|}^{p^{*}}-|u_n-{ũ}_n{|}^{p^{*}}-|{ũ}_n{|}^{p^{*}}\right)+\left(|v_n{|}^{p^{*}}-|v_n-{ṽ}_n{|}^{p^{*}}-|{ṽ}_n{|}^{p^{*}}\right)\right)=0$$

and

$$\underset{n\to \infty }{\text{lim}}\underset{{ℝ}^N}{\int }\left(H\left(u_n,v_n\right)-H\left(u_n-{ũ}_n,v_n{ṽ}_n\right)-H\left({ũ}_n,{ṽ}_n\right)\right)=0.$$

In connection with the fact I _λ (u _n , v _n ) → c and $I_{\lambda }\left({ũ}_n,{ṽ}_n\right)\to I_{\lambda }\left(u,v\right)$, we obtain

$$I_{\lambda }\left(u_n-{ũ}_n,v_n-{ṽ}_n\right)\to c-I_{\lambda }\left(u,v\right).$$

In the following, we will verify the fact $I_{\lambda }^{\prime }\left(u_n-{ũ}_n,v_n-{ṽ}_n\right)\to 0$.

For any (φ, ψ) ∈ B, it follows that

$$\begin{array}{l}{I}_{\lambda }^{\text{'}}(u_n-{\tilde{u}}_n,v_n-{\tilde{v}}_n)(\phi ,\psi )\\ =I_{\lambda }^{\text{'}}(u_n,v_n)(\phi ,\psi )-I_{\lambda }^{\text{'}}({\tilde{u}}_n,{\tilde{v}}_n)(\phi ,\psi )\\ +\lambda {\displaystyle \underset{{ℝ}^N}{\int }K(x)[(|u_n{|}^{p^{*}-2}{u}_n-|u_n-{\tilde{u}}_n{|}^{p^{*}-2}(u_n-{\tilde{u}}_n)-|{\tilde{u}}_n{|}^{p^{*}-2}{\tilde{u}}_n)\phi }\\ +(|v_n{|}^{p^{*}-2}{v}_n-|v_n-{\tilde{v}}_n{|}^{p^{*}-2}(v_n-{\tilde{v}}_n)-|{\tilde{v}}_n{|}^{p^{*}-2}{\tilde{v}}_n)\psi ]\\ +\lambda {\displaystyle \underset{{ℝ}^N}{\int }[(H_s(u_n,v_n)-H_s(u_n-{\tilde{u}}_n,v_n-{\tilde{v}}_n)-H_s({\tilde{u}}_n,{\tilde{v}}_n))\phi }\\ +(H_t(u_n,v_n)-H_t(u_n-{\tilde{u}}_n,v_n-{\tilde{v}}_n)-H_t({\tilde{u}}_n,{\tilde{v}}_n))\psi ]+o(1).\end{array}$$

Standard argument shows that

$$\underset{n\to \infty }{\text{lim}}\underset{{ℝ}^N}{\int }K\left(x\right)\left(|u_n{|}^{p^{*}-2}{u}_n-|u_n-{ũ}_n{|}^{p^{*}-2}\left(u_n-{ũ}_n\right)-|{ũ}_n{|}^{p^{*}-2}{ũ}_n\right)\phi =0$$

and

$$\underset{n\to \infty }{\text{lim}}\underset{{ℝ}^N}{\int }K\left(x\right)\left(|v_n{|}^{p^{*}-2}{v}_n-|v_n-{ṽ}_n{|}^{p^{*}-2}\left(v_n-{ṽ}_n\right)-|{ṽ}_n{|}^{p^{*}-2}{ṽ}_n\right)\psi =0$$

uniformly in ‖φ, ψ)‖ _B ≤ 1.

By Lemma 3.3, we have

$$\underset{n\to \infty }{\text{lim}}\underset{{ℝ}^N}{\int }\left(H_s\left(u_n,v_n\right)-H_s\left(u_n-{ũ}_n,v_n-{ṽ}_n\right)-H_s\left({ũ}_n,{ṽ}_n\right)\right)\phi =0$$

and

$$\underset{n\to \infty }{\text{lim}}\underset{{ℝ}^N}{\int }\left(H_t\left(u_n,v_n\right)-H_t\left(u_n-{ũ}_n,v_n-{ṽ}_n\right)-H_t\left({ũ}_n,{ṽ}_n\right)\right)\psi =0$$

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

$$I_{\lambda }^{\prime }\left(u_n-{ũ}_n,v_n-{ṽ}_n\right)\to 0\text{in }{B}^{-1}.$$

□

Let $u_n^1=u_n-{ũ}_n$, $v_n^1=v_n-{ṽ}_n$, then $u_n-u=u_n^1+\left({ũ}_n-u\right)$, $v_n-v=v_n^1+\left({ṽ}_n-v\right)$. From (3.1), we get (u _n , v _n ) → (u, v) in B if and only if $\left(u_n^1,v_n^1\right)\to \left(0,0\right)$ in B.

Observe that

$$\begin{array}{l}{I}_{\lambda }\left(u_n^1,v_n^1\right)-\frac{1}{p}{I}_{\lambda }^{\text{'}}\left(u_n^1,v_n^1\right)\left(u_n^1,v_n^1\right)\\ =\left(\frac{1}{p}-\frac{1}{p^{*}}\right)\lambda \underset{{ℝ}^N}{\int }K\left(x\right)\left(|u_n^1{|}^{p^{*}}+|v_n^1{|}^{p^{*}}\right)\\ +\lambda \underset{{ℝ}^N}{\int }\left(\frac{1}{p}\left(u_n^1{H}_s\left(u_n^1,v_n^1\right)+v_n^1{H}_t\left(u_n^1,v_n^1\right)\right)-H\left(u_n^1,v_n^1\right)\right)\\ \ge \frac{\lambda }{N}\underset{{ℝ}^N}{\int }K\left(x\right)\left(|u_n^1{|}^{p^{*}}+|v_n^1{|}^{p^{*}}\right)\\ \ge \frac{\lambda }{N}{K}_{\text{min}}\underset{{ℝ}^N}{\int }\left(|u_n^1{|}^{p^{*}}+|v_n^1{|}^{p^{*}}\right),\end{array}$$

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

Thus by Lemma 3.4, we get

$$\left|\right|\left(u_n^1,v_n^1\right)|{|}_{p^{*}}^{p^{*}}\le \frac{N\left(c-I_{\lambda }\left(u,v\right)\right)}{\lambda K_{\text{min}}}+o\left(1\right).$$

(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 $L_{\text{loc}}^p\left({ℝ}^N\right)$, we get

$$\underset{{ℝ}^N}{\int }V\left(x\right)\left(|u_n^1{|}^p+|v_n^1{|}^p\right)=\underset{{ℝ}^N}{\int }{V}_b\left(x\right)\left(|u_n^1{|}^p+|v_n^1{|}^p\right)+o\left(1\right).$$

(3.4)

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

$$\begin{array}{l}\underset{{ℝ}^N}{\int }\left(K\left(x\right)\left(|u{|}^{p^{*}}+|v{|}^{p^{*}}\right)+u{H}_s\left(u,v\right)+v{H}_t\left(u,v\right)\right)\\ \hfill \le & b\left(\left|\right|u|{|}_p^p+|\left|v\right|{|}_p^p\right)+C_b\left(\left|\right|u|{|}_{p^{*}}^{p^{*}}+|\left|v\right|{|}_{p^{*}}^{p^{*}}\right).\end{array}$$

(3.5)

Let S be the best Sobolev constant of the immersion

$$S\left|\right|u|{|}_{p^{*}}^p\le \underset{{ℝ}^N}{\int }|\nabla u{|}^p\text{for all }u\in W^{1,p}\left({ℝ}^N\right).$$

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 $c-I_{\lambda }\left(u,v\right)\ge {\alpha }_0{\lambda }^{1-\frac{N}{p}}$.

Proof. Assume that (u _n , v _n ) ↛ (u, v), then

$$\lim \underset{n\to \infty }{\inf }\left|\right|(u_n^1,v_n^1)|{|}_{\lambda }>0$$

and

$$c-J_{\lambda }\left(u,v\right)>0.$$

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

$$\begin{array}{l}S\left(\left|\right|u_n^1|{|}_{p^{*}}^p+|\left|v_n^1\right|{|}_{p^{*}}^p\right)\\ \le \underset{{ℝ}^N}{\int }\left(|\nabla u_n^1{|}^p+|\nabla v_n^1{|}^p\right)\\ =\underset{{ℝ}^N}{\int }\left(|\nabla u_n^1{|}^p+\lambda V\left(x\right)|u_n^1{|}^p+|\nabla v_n^1{|}^p+\lambda V\left(x\right)|v_n^1{|}^p\right)-\lambda \underset{{ℝ}^N}{\int }V\left(x\right)\left(|u_n^1{|}^p+|v_n^1{|}^p\right)\\ =\lambda \underset{{ℝ}^N}{\int }K\left(x\right)\left(|u_n^1{|}^{p^{*}}+|v_n^1{|}^{p^{*}}\right)+u_n^1{H}_s\left(u_n^1,v_n^1\right)+v_n^1{H}_t\left(u_n^1,v_n^1\right)\\ -\lambda \underset{{ℝ}^N}{\int }V\left(x\right)\left(|u_n^1{|}^p+|v_n^1{|}^p\right)+o\left(1\right)\\ \le \lambda b\left(\left|\right|u_n^1|{|}_p^p+|\left|v_n^1\right|{|}_p^p\right)+\lambda C_b\left(\left|\right|u_n^1|{|}_{p^{*}}^{p^{*}}+|\left|v_n^1\right|{|}_{p^{*}}^{p^{*}}\right)-\lambda b\left(\left|\right|u_n^1|{|}_p^p+|\left|v_n^1\right|{|}_p^p\right)+o\left(1\right)\\ =\lambda C_b\left(\left|\right|u_n^1|{|}_{p^{*}}^{p^{*}}+|\left|v_n^1\right|{|}_{p^{*}}^{p^{*}}\right)+o\left(1\right).\end{array}$$

This, together with $\lim {\inf }_{n\to \infty }(|\left|u_n^1\right|{|}_{p^{*}}^{p^{*}}+\left|\right|v_n^1|{|}_{p^{*}}^{p^{*}})>0$ and (3.3), gives

$$\begin{array}{ll}\hfill S& \le \lambda C_b{\left(\left|\right|u_n^1|{|}_{p^{*}}^{p^{*}}+|\left|v_n^1\right|{|}_{p^{*}}^{p^{*}}\right)}^{\frac{p^{*}-p}{p^{*}}}+o\left(1\right)\\ \le \lambda C_b{\left(\frac{N\left(c-I_{\lambda }\left(u,v\right)\right)}{\lambda K_{\text{min}}}\right)}^{\frac{p}{N}}+o\left(1\right)\\ ={\lambda }^{1-\frac{p}{N}}{C}_b{\left(\frac{N}{K_{\text{min}}}\right)}^{\frac{p}{N}}{\left(c-I_{\lambda }\left(u,v\right)\right)}^{\frac{p}{N}}+o\left(1\right).\end{array}$$

Set ${\alpha }_0=S^{\frac{N}{p}}{C}_b^{-\frac{N}{p}}{N}^{-1}{K}_{\text{min}}$, then

$${\alpha }_0{\lambda }^{1-\frac{N}{p}}\le c-I_{\lambda }\left(u,v\right)+o\left(1\right).$$

This proof is completed. □

Since $W^{1,p}\left({ℝ}^N\right)\hookrightarrow L^{p^{*}}\left({ℝ}^N\right)$ 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 $c\le {\alpha }_0{\lambda }^{1-\frac{N}{p}}$, the sequence {(u _n , v _n )} has a strongly convergent subsequence in B.

Proof. By the fact $c\le {\alpha }_0{\lambda }^{1-\frac{N}{p}}$, we have

$$c-I_{\lambda }\left(u,v\right)\le {\alpha }_0{\lambda }^{1-\frac{N}{p}}-I_{\lambda }\left(u,v\right).$$

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

$$I_{\lambda }\left(u,v\right)>0\text{if}0<\left|\right|\left(u,v\right)|{|}_{\lambda }<{\rho }_{\lambda }\text{and }{I}_{\lambda }\left(u,v\right)\ge {\alpha }_{\lambda }\text{if}|\left|\left(u,v\right)\right|{|}_{\lambda }={\rho }_{\lambda }.$$

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

$$\underset{{ℝ}^N}{\int }G\left(u,v\right)\le \delta \left(\left|\right|u|{|}_p^p+|\left|v\right|{|}_p^p\right)+C_{\delta }\left(\left|\right|u|{|}_{p^{*}}^{p^{*}}+|\left|v\right|{|}_{p^{*}}^{p^{*}}\right).$$

Thus

$$\begin{array}{ll}\hfill I_{\lambda }\left(u,v\right)& =\frac{1}{p}\left|\right|\left(u,v\right)|{|}_{\lambda }^p-\lambda \underset{{ℝ}^N}{\int }G\left(u,v\right)\\ \ge \frac{1}{p}\left|\right|\left(u,v\right)|{|}_{\lambda }^p-\lambda \delta \left(\left|\right|u|{|}_p^p+|\left|v\right|{|}_p^p\right)-\lambda C_{\delta }\left(\left|\right|u|{|}_{p^{*}}^{p^{*}}+|\left|v\right|{|}_{p^{*}}^{p^{*}}\right).\end{array}$$

Note that $\left|\right|u|{|}_p^p+|\left|v\right|{|}_p^p\le C_1\left|\right|\left(u,v\right)|{|}_{\lambda }^p$. If δ ≤ (2pλC₁)^-1, then

$$I_{\lambda }\left(u,v\right)\ge \frac{1}{2p}\left|\right|\left(u,v\right)|{|}_{\lambda }^p-\lambda C_{\delta }\left(\left|\right|u|{|}_{p^{*}}^{p^{*}}+|\left|v\right|{|}_{p^{*}}^{p^{*}}\right).$$

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

$$I_{\lambda }\left(u,v\right)\to -\infty \text{as }\left(u,v\right)\in F,\left|\right|\left(u,v\right)|{|}_{\lambda }\to \infty .$$

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

$$I_{\lambda }\left(u,v\right)\le \frac{1}{p}\left|\right|\left(u,v\right)|{|}_{\lambda }^p-\lambda a_0\left(|u{|}_{\alpha }^{\alpha }+|v{|}_{\beta }^{\beta }\right)\text{for}\text{all }\left(u,v\right)\in B.$$

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

$${\Phi }_{\lambda }\left(u,v\right)=\frac{1}{p}\underset{{ℝ}^N}{\int }\left(|\nabla u{|}^p+\lambda V\left(x\right)|u{|}^p+|\nabla v{|}^p+\lambda V\left(x\right)|v{|}^p\right)-\lambda a_0\underset{{ℝ}^N}{\int }\left(|u{|}^{\alpha }+|v{|}^{\beta }\right).$$

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

Observe that

$$\text{inf}\left\{\underset{{ℝ}^N}{\int }|\nabla \varphi {|}^p:\varphi \in C_0^{\infty }\left({ℝ}^N,ℝ\right),|\varphi {|}_{L^{\alpha }\left({ℝ}^N\right)}=1\right\}=0$$

and

$$\text{inf}\left\{\underset{{ℝ}^N}{\int }|\nabla \psi {|}^p:\psi \in C_0^{\infty }\left({ℝ}^N,ℝ\right),|\psi {|}_{L^{\beta }\left({ℝ}^N\right)}=1\right\}=0.$$

For any δ> 0, there are φ _δ , ${\psi }_{\delta }\in C_0^{\infty }\left({ℝ}^N,ℝ\right)$ with $|{\varphi }_{\delta }{|}_{L^{\alpha }\left({ℝ}^N\right)}=|{\psi }_{\delta }{|}_{L^{\beta }\left({ℝ}^N\right)}=1$ and suppφ_δ, $\text{supp}{\psi }_{\delta }\subset B_{r_{\delta }}\left(0\right)$ such that $|\nabla {\varphi }_{\delta }{|}_p^p,|\nabla {\psi }_{\delta }{|}_p^p<\delta$.

Let $w_{\lambda }\left(x\right)=\left({\varphi }_{\delta }\left(\sqrt[p]{\lambda }x\right),{\psi }_{\delta }\left(\sqrt[p]{\lambda }x\right)\right)$, then $\text{supp}{w}_{\lambda }\subset B_{{\lambda }^{-\frac{1}{p}{r}_{\delta }}}\left(0\right)$. For t ≥ 0, we get

$$\begin{array}{ll}\hfill {\Phi }_{\lambda }\left(t{w}_{\lambda }\right)& =\frac{t^p}{p}{∥w_{\lambda }∥}_{\lambda }^p-a_0\lambda t^{\alpha }{\int }_{{ℝ}^N}|{\varphi }_{\delta }\left(\sqrt[p]{\lambda }x\right){|}^{\alpha }-a_0\lambda t^{\beta }\underset{{ℝ}^N}{\int }|{\psi }_{\delta }\left(\sqrt[p]{\lambda }x\right){|}^{\beta }\\ ={\lambda }^{1-\frac{N}{p}}{J}_{\lambda }\left(t{\varphi }_{\delta },t{\psi }_{\delta }\right),\end{array}$$

where

$$J_{\lambda }\left(u,v\right)=\frac{1}{p}\underset{{ℝ}^N}{\int }\left(|\nabla u{|}^p+|\nabla v{|}^p+V\left({\lambda }^{-\frac{1}{p}}x\right)\left(|u{|}^p+|v{|}^p\right)\right)-a_0\underset{{ℝ}^N}{\int }\left(|u{|}^{\alpha }+|v{|}^{\beta }\right).$$

We easily prove that

$$\begin{array}{c}\underset{t\ge 0}{\max }{J}_{\lambda }(t{\varphi }_{\delta },t{\psi }_{\delta })\le \frac{\alpha -p}{p\alpha {(\alpha a_0)}^{\frac{p}{\alpha -p}}}{\left\{{\displaystyle \underset{{ℝ}^N}{\int }(|\nabla {\varphi }_{\delta }{|}^p+V({\lambda }^{-\frac{1}{p}}x)|{\varphi }_{\delta }{|}^p}\right\}}^{\frac{\alpha }{\alpha -p}}\\ +\frac{\beta -p}{p\beta {(\beta a_0)}^{\frac{p}{\beta -p}}}{\left\{{\displaystyle \underset{{ℝ}^N}{\int }(|\nabla {\psi }_{\delta }{|}^p+V({\lambda }^{-\frac{1}{p}}x)|{\psi }_{\delta }{|}^p}\right\}}^{\frac{\beta }{\beta -p}}.\end{array}$$

Together with V (0) = 0 and $|\nabla {\varphi }_{\delta }{|}_p^p,|\nabla {\psi }_{\delta }{|}_p^p<\delta$, this implies that there is Λ_δ> 0 such that for all λ ≥ Λ_δ, we have

$$\underset{t\ge 0}{\text{max}}{I}_{\lambda }\left(t{\varphi }_{\delta },t{\psi }_{\delta }\right)\le \left(\frac{\alpha -p}{p\alpha {\left(\alpha a_0\right)}^{\frac{p}{\alpha -p}}}{\left(2\delta \right)}^{\frac{\alpha }{\alpha -p}}+\frac{\beta -p}{p\beta {\left(\beta a_0\right)}^{\frac{p}{\beta -p}}}{\left(2\delta \right)}^{\frac{\beta }{\beta -p}}\right){\lambda }^{1-\frac{N}{p}}.$$

(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 ${\bar{w}}_{\lambda }\in B$ with ${∥{\bar{w}}_{\lambda }∥}_{\lambda }>{\rho }_{\lambda },I_{\lambda }\left({\bar{w}}_{\lambda }\right)\le 0$ and

$$\underset{t\ge 0}{\text{max}}{I}_{\lambda }\left(t{\bar{w}}_{\lambda }\right)\le \sigma {\lambda }^{1-\frac{N}{p}},$$

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

$$c_{\lambda }=\underset{\gamma \in {\Gamma }_{\lambda }}{\text{inf}}\underset{t\in \left[0,1\right]}{\text{max}}{I}_{\lambda }\left(\gamma \left(t\right)\right)\le \sigma {\lambda }^{1-\frac{N}{p}},$$

where ${\Gamma }_{\lambda }=\left\{\gamma \in C\left(\left[0,1\right],B\right):\gamma \left(0\right)=0,\gamma \left(1\right)={\bar{w}}_{\lambda }\right\}.$

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 $I_{\lambda }^{\prime }\left(u_{\lambda },v_{\lambda }\right)=0.$ 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 $I_{\lambda }\left(u_{\lambda },v_{\lambda }\right)\le \sigma {\lambda }^{1-\frac{N}{p}}$ and $I_{\lambda }^{\prime }\left(u_{\lambda },v_{\lambda }\right)=0.$ we have

$$\begin{array}{ll}\hfill I_{\lambda }\left(u_{\lambda },v_{\lambda }\right)& =I_{\lambda }\left(u_{\lambda },v_{\lambda }\right)-\frac{1}{\theta }{I}_{\lambda }^{\text{'}}\left(u_{\lambda },v_{\lambda }\right)\left(u_{\lambda },v_{\lambda }\right)\\ =\left(\frac{1}{p}-\frac{1}{\theta }\right){∥\left(u_{\lambda },v_{\lambda }\right)∥}_{\lambda }^p+\left(\frac{1}{\theta }-\frac{1}{p^{*}}\right)\lambda \underset{{ℝ}^N}{\int }K\left(x\right)\left(|u_{\lambda }{|}^{p^{*}}+|v_{\lambda }{|}^{p^{*}}\right)\\ +\lambda \underset{{ℝ}^N}{\int }\left(\frac{1}{\theta }\left(u_{\lambda }{H}_s\left(u_{\lambda },v_{\lambda }\right)+v_{\lambda }{H}_t\left(u_{\lambda },v_{\lambda }\right)\right)-H\left(u_{\lambda },v_{\lambda }\right)\right)\\ \ge \left(\frac{1}{p}-\frac{1}{\theta }\right){∥\left(u_{\lambda },v_{\lambda }\right)∥}_{\lambda }^p.\end{array}$$

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