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

## Abstract

We consider a p-Laplacian system with critical nonlinearity in N . Under the proper assumptions, we obtain the existence of nontrivial solutions to perturbed p-Laplacian system by using the variational approach.

MR Subject Classification: 35B33; 35J60; 35J65.

## 1 Introduction

$\left\{\begin{array}{l}-{\epsilon }^{p}{\Delta }_{p}u+V\left(x\right)|u{|}^{p-2}u=K\left(x\right)|u{|}^{{p}^{*}-2}u+{H}_{u}\left(u,v\right),\phantom{\rule{0.5em}{0ex}}x\in {ℝ}^{N},\hfill \\ -{\epsilon }^{p}{\Delta }_{p}v+V\left(x\right)|v{|}^{p-2}v=K\left(x\right)|v{|}^{{p}^{*}-2}v+{H}_{v}\left(u,v\right),\phantom{\rule{0.5em}{0ex}}x\in {ℝ}^{N},\hfill \\ u\left(x\right),\phantom{\rule{0.5em}{0ex}}v\left(x\right)>0,\hfill \\ u\left(x\right),\phantom{\rule{0.5em}{0ex}}v\left(x\right)\to 0\phantom{\rule{0.5em}{0ex}}\text{as}|x|\to \infty ,\hfill \end{array}$
(1.1)

where Δ p u = div(|u|p-2u) is the p-Laplacian operator, 1 < p < N and p* = Np/(N − p) is the critical exponent.

Throughout the article, we will assume that:

(V0) V C( N ), V (0) = inf V (x) = 0 and there exists b > 0 such that the set νb := {x N : V (x) < b} has finite Lebesgue measure;

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

(H1) H C1(2) and H s , H t = o(|s|p-1+ |t|p-1) as |s| + |t| → 0;

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

$|{H}_{s}\left(s,\phantom{\rule{2.77695pt}{0ex}}t\right)|,\phantom{\rule{2.77695pt}{0ex}}|{H}_{t}\left(s,\phantom{\rule{2.77695pt}{0ex}}t\right)|\phantom{\rule{0.3em}{0ex}}\le c\left(1+|s{|}^{q-1}+|t{|}^{q-1}\right);$

(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

$-{\epsilon }^{p}{\Delta }_{p}u+V\left(x\right)|u{|}^{p-2}u=K\left(x\right)|u{|}^{{p}^{*}-2}u+h\left(x,\phantom{\rule{2.77695pt}{0ex}}u\right),\phantom{\rule{1em}{0ex}}x\in {ℝ}^{N}.$
(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 , Ambrosetti  and Guedda and Veron  (see also their references) in bounded domains. As far as unbounded domains are concerned, we recall the work by Benci and Cerami , Floer and Weistein , Oh , Clapp , Del Pino and Felmer , Cingolani and Lazzo , Ding and Lin . Especially, in , the authors studied the Equation (1.2) in the case p = 2. In that article, they made the following assumptions:

(A1) V C( N ), min V = 0 and there is b > 0 such that the set νb := {x N : V (x) < b} has finite Lebesgue measure;

(A2) K(x) C( N ), 0 < inf K ≤ sup K <

(B1) h C( N × ) and h(x, u) = o(|u|) uniformly in x as |u| → 0;

(B2) there are c0> 0, q < 2* such that |h(x, u)| ≤ c0(1 + |u|q-1) for all (x, u);

(B3) there are a0> 0, p > 2 and µ > 2 such that H(x, u) = a0|u|p and µH(x, u) ≤ h(x, u)u for all (x, u), where $H\left(x,u\right)={\int }_{0}^{u}h\left(x,s\right)ds$.

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  considered (1.2) with ε ≤ 1, V (x) = µ, K(x) = 1, h(x, u) = 0 and proved that (1.2) has a solution if p2N and µ (0, λ1), where λ1 is the first eigenvalue of the p-Laplacian. In , Alves and Ding studied the same problem of  and obtained the multiplicity of positive solutions in bounded domain Ω N . Moreover, Liu and Zheng  investigated (1.2) in N with ε = 1 and K(x) = 0. Under the sign-changing potential and subcritical p-superlinear nonlinearity, the authors got the existence result.

Motivated by some results found in [10, 11, 13], a natural question arises whether existence of nontrivial solutions continues to hold for the p-Laplacian system with the critical nonlinearity in N .

The main difficulty in the case above mentioned is the lack of compactness of the energy functional associated to the system (1.1) because of unbounded domain N and critical nonlinearity. To overcome this difficulty, we make careful estimates and prove that there is a Palais-Smale sequence that has a strongly convergent sequence. The method or idea here is similar to the one of . 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 .

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 D1,p( N ) be the completion of ${C}_{0}^{\infty }\left({ℝ}^{N}\right)$ under

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

We introduce the space

$E\left({ℝ}^{N},\phantom{\rule{2.77695pt}{0ex}}V\right)=\left\{u\in {W}^{1,p}\left({ℝ}^{N}\right):\underset{{ℝ}^{N}}{\int }V\left(x\right)|u{|}^{p}<\infty \right\}$

equipped with the norm

$||u|{|}_{E}={\left(\underset{{ℝ}^{N}}{\int }\left(|\nabla u{|}^{p}+\phantom{\rule{0.3em}{0ex}}V\left(x\right)|u{|}^{p}\right)\right)}^{\frac{1}{p}}$

and the space

${E}_{\lambda }\left({ℝ}^{N},\phantom{\rule{2.77695pt}{0ex}}V\right)=\left\{u\in {W}^{1,p}\left({ℝ}^{N}\right):\underset{{ℝ}^{N}}{\int }\lambda V\left(x\right)|u{|}^{p}<\infty ,\phantom{\rule{1em}{0ex}}\lambda >0\right\}$

under

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

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

Set B = E λ × E λ and $||\left(u,v\right)|{|}_{\lambda }=\phantom{\rule{0.3em}{0ex}}||u|{|}_{\lambda }^{p}+||v|{|}_{\lambda }^{p}$ for any (u, v) B. Let λ = ε-pin the system (1.1), then (1.1) is changed into

$\left\{\begin{array}{l}-{\Delta }_{p}u+\lambda V\left(x\right)|u{|}^{p-2}u=\lambda K\left(x\right)|u{|}^{{p}^{*}-2}u+\lambda {H}_{u}\left(u,v\right),\phantom{\rule{0.5em}{0ex}}\in {ℝ}^{N},\hfill \\ -{\Delta }_{p}v+\lambda V\left(x\right)|v{|}^{p-2}v=\lambda K\left(x\right)|v{|}^{{p}^{*}-2}v+\lambda {H}_{v}\left(u,v\right),\phantom{\rule{0.5em}{0ex}}x\in {ℝ}^{N},\hfill \\ u\left(x\right),\phantom{\rule{0.5em}{0ex}}v\left(x\right)>0,\hfill \\ u\left(x\right),\phantom{\rule{0.5em}{0ex}}v\left(x\right)\to 0,\phantom{\rule{0.5em}{0ex}}\text{as}\phantom{\rule{0.5em}{0ex}}|x|\to \infty .\hfill \end{array}$
(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

$\begin{array}{ll}\hfill {I}_{\lambda }\left(u,\phantom{\rule{2.77695pt}{0ex}}v\right)& =\frac{1}{p}\underset{{ℝ}^{N}}{\int }\left(|\nabla u{|}^{p}+\lambda V\left(x\right)|u{|}^{p}+|\nabla v{|}^{p}+\lambda V\left(x\right)|v{|}^{p}\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}-\frac{\lambda }{{p}^{*}}\underset{{ℝ}^{N}}{\int }K\left(x\right)\left(|u{|}^{{p}^{*}}+|v{|}^{{p}^{*}}\right)-\lambda \underset{{ℝ}^{N}}{\int }H\left(u,\phantom{\rule{2.77695pt}{0ex}}v\right)\phantom{\rule{2em}{0ex}}\\ =\frac{1}{p}||\left(u,\phantom{\rule{2.77695pt}{0ex}}v\right)|{|}_{\lambda }^{p}-\lambda \underset{{ℝ}^{N}}{\int }G\left(u,\phantom{\rule{2.77695pt}{0ex}}v\right),\phantom{\rule{2em}{0ex}}\end{array}$

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  show that I λ C1(B, ) and its critical points are the weak solutions of (2.1).

## 3 Technical lemmas

In this section, we will recall and prove some lemmas which are crucial in the proof of the main result.

Lemma 3.1. Let the assumptions of Theorem 2 be satisfied. If the sequence {(u n , v n )} B is a (PS) c sequence for I λ , then we get that c ≥ 0 and {(u n , v n )} is bounded in the space B.

Proof. One has

$\begin{array}{l}\phantom{\rule{1em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{I}_{\lambda }\left({u}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}\right)-\frac{1}{\theta }{I}_{\lambda }^{\text{'}}\left({u}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}\right)\left({u}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}\right)\phantom{\rule{2em}{0ex}}\\ =\frac{1}{p}||\left({u}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}\right)|{|}_{\lambda }^{p}-\frac{\lambda }{{p}^{*}}\underset{{ℝ}^{N}}{\int }K\left(x\right)\left(|{u}_{n}{|}^{{p}^{*}}+|{v}_{n}{|}^{{p}^{*}}\right)-\lambda \underset{{ℝ}^{N}}{\int }H\left({u}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}-\frac{1}{\theta }\left[||\left({u}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}\right)|{|}_{\lambda }^{p}-\lambda \underset{{ℝ}^{N}}{\int }K\left(x\right)\left(|{u}_{n}{|}^{{p}^{*}}+|{v}_{n}{|}^{{p}^{*}}\right)-\lambda \underset{{ℝ}^{N}}{\int }\left({u}_{n}{H}_{s}\left({u}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}\right)+{v}_{n}{H}_{t}\left({u}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}\right)\right)\right]\phantom{\rule{2em}{0ex}}\\ =\left(\frac{1}{p}-\frac{1}{\theta }\right)||\left({u}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}\right)|{|}_{\lambda }^{p}+\left(\frac{1}{\theta }-\frac{1}{{p}^{*}}\right)\lambda \underset{{ℝ}^{N}}{\int }K\left(x\right)\left(|{u}_{n}{|}^{{p}^{*}}+|{v}_{n}{|}^{{p}^{*}}\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+\lambda \underset{{ℝ}^{N}}{\int }\left(\frac{1}{\theta }\left({u}_{n}{H}_{s}\left({u}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}\right)+{v}_{n}{H}_{t}\left({u}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}\right)\right)-H\left({u}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}\right)\right)\phantom{\rule{2em}{0ex}}\end{array}$

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

${I}_{\lambda }\left({u}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}\right)-\frac{1}{\theta }{I}_{\lambda }^{\prime }\left({u}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}\right)\left({u}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}\right)\ge \left(\frac{1}{p}-\frac{1}{\theta }\right)||\left({u}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}\right)|{|}_{\lambda }^{p}.$

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 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

$\underset{i\to \infty }{\text{lim}}\text{sup}\underset{{B}_{i}\{B}_{r}}{\int }\left(|{u}_{{n}_{i}}{|}^{d}+|{v}_{{n}_{i}}{|}^{d}\right)\le \epsilon$

for any r ≥ r ε , where B r := {x N : |x| ≤ r}.

Proof. The proof of Lemma 3.2 is similar to the one of Lemma 3.2 of , so we omit it. □

Let η C(+) be a smooth function satisfying 0 ≤ η(t) 1, η(t) = 1 if t ≤ 1 and η(t) = 0 if t ≥ 2. Define ${ũ}_{j}\left(x\right)=\eta \left(2|x|/j\right)u\left(x\right)$, ${ṽ}_{j}\left(x\right)=\eta \left(2|x|/j\right)v\left(x\right)$. It is obvious that

(3.1)

Lemma 3.3. One has

$\underset{j\to \infty }{\text{lim}}\underset{{ℝ}^{N}}{\int }\left({H}_{s}\left({u}_{{n}_{j}},\phantom{\rule{2.77695pt}{0ex}}{v}_{{n}_{j}}\right)-{H}_{s}\left({u}_{{n}_{j}}-{ũ}_{j},\phantom{\rule{2.77695pt}{0ex}}{v}_{{n}_{j}}-{ṽ}_{j}\right)-{H}_{s}\left({ũ}_{j},\phantom{\rule{2.77695pt}{0ex}}{ṽ}_{j}\right)\right)\phi =0$

and

$\underset{j\to \infty }{\text{lim}}\underset{{ℝ}^{N}}{\int }\left({H}_{t}\left({u}_{{n}_{j}},\phantom{\rule{2.77695pt}{0ex}}{v}_{{n}_{j}}\right)-{H}_{t}\left({u}_{{n}_{j}}-{ũ}_{j},\phantom{\rule{2.77695pt}{0ex}}{v}_{{n}_{j}}-{v}_{j}\right)-{H}_{t}\left({ũ}_{j},\phantom{\rule{2.77695pt}{0ex}}{ṽ}_{j}\right)\right)\psi =0$

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}\phantom{\rule{0.5em}{0ex}}\underset{j\to \infty }{\mathrm{lim}}\mathrm{sup}\underset{{ℝ}^{N}}{\int }\left({H}_{s}\left({u}_{{n}_{j}},\phantom{\rule{0.5em}{0ex}}{v}_{{n}_{j}}\right)-{H}_{s}\left({u}_{{n}_{j}}-{\stackrel{˜}{u}}_{j},\phantom{\rule{0.5em}{0ex}}{v}_{{n}_{j}}-{\stackrel{˜}{v}}_{j}\right)-{H}_{s}\left({\stackrel{˜}{u}}_{j},\phantom{\rule{0.5em}{0ex}}{\stackrel{˜}{v}}_{j}\right)\right)\phi \\ =\underset{j\to \infty }{\mathrm{lim}}\mathrm{sup}\underset{{B}_{j}}{\int }\left({H}_{s}\left({u}_{{n}_{j}},\phantom{\rule{0.5em}{0ex}}{v}_{{n}_{j}}\right)-{H}_{s}\left({u}_{{n}_{j}}-{\stackrel{˜}{u}}_{j},\phantom{\rule{0.5em}{0ex}}{v}_{{n}_{j}}-{\stackrel{˜}{v}}_{j}\right)-{H}_{s}\left({\stackrel{˜}{u}}_{j},\phantom{\rule{0.5em}{0ex}}{\stackrel{˜}{v}}_{j}\right)\right)\phi \\ =\underset{j\to \infty }{\mathrm{lim}}\mathrm{sup}\underset{{B}_{j}\{B}_{r}}{\int }\left({H}_{s}\left({u}_{{n}_{j}},\phantom{\rule{0.5em}{0ex}}{v}_{{n}_{j}}\right)-{H}_{s}\left({u}_{{n}_{j}}-{\stackrel{˜}{u}}_{j},\phantom{\rule{0.5em}{0ex}}{v}_{{n}_{j}}-{\stackrel{˜}{v}}_{j}\right)-{H}_{s}\left({\stackrel{˜}{u}}_{j},\phantom{\rule{0.5em}{0ex}}{\stackrel{˜}{v}}_{j}\right)\right)\phi \\ \le c\underset{j\to \infty }{\mathrm{lim}}\mathrm{sup}\underset{{B}_{j}\{B}_{r}}{\int }\left(|{u}_{{n}_{j}}{|}^{p-1}+|{v}_{{n}_{j}}{|}^{p-1}+|{u}_{{n}_{j}}{|}^{q-1}+|{v}_{{n}_{j}}{|}^{q-1}+|{\stackrel{˜}{u}}_{j}{|}^{p-1}+|{\stackrel{˜}{v}}_{j}{|}^{p-1}\\ \phantom{\rule{0.5em}{0ex}}+|{\stackrel{˜}{u}}_{j}{|}^{q-1}+|{\stackrel{˜}{v}}_{j}{|}^{q-1}+|{u}_{{n}_{j}}-{\stackrel{˜}{u}}_{j}{|}^{p-1}+|{v}_{{n}_{j}}-{\stackrel{˜}{v}}_{j}{|}^{p-1}+|{u}_{{n}_{j}}-{\stackrel{˜}{u}}_{j}{|}^{q-1}+|{v}_{{n}_{j}}-{\stackrel{˜}{v}}_{j}{|}^{q-1}\right)\phi \\ \le {c}_{1}\underset{j\to \infty }{\mathrm{lim}}\mathrm{sup}\underset{{B}_{j}\{B}_{r}}{\int }\left(|{u}_{{n}_{j}}{|}^{p-1}+|{v}_{{n}_{j}}{|}^{p-1}+|{\stackrel{˜}{u}}_{j}{|}^{p-1}+|{\stackrel{˜}{v}}_{j}{|}^{p-1}\right)\phi \\ \phantom{\rule{0.5em}{0ex}}+{c}_{2}\underset{j\to \infty }{\mathrm{lim}}\mathrm{sup}\underset{{B}_{j}\{B}_{r}}{\int }\left(|{u}_{{n}_{j}}{|}^{q-1}+|{v}_{{n}_{j}}{|}^{q-1}+|{\stackrel{˜}{u}}_{j}{|}^{q-1}+|{\stackrel{˜}{v}}_{j}{|}^{q-1}\right)\phi \end{array}$
(3.2)

By Hölder inequality and Lemma 3.2, it follows that

$\begin{array}{ll}\hfill \underset{j\to \infty }{\text{lim}}\text{sup}\underset{{B}_{j}\{B}_{r}}{\int }|{u}_{{n}_{j}}{|}^{p-1}|\phi |& \le \underset{j\to \infty }{\text{lim}}\text{sup}{\left(\underset{{B}_{j}\{B}_{r}}{\int }|{u}_{{n}_{j}}{|}^{p}\right)}^{\frac{p-1}{p}}{\left(\underset{{B}_{j}\{B}_{r}}{\int }|\phi {|}^{p}\right)}^{\frac{1}{p}}\phantom{\rule{2em}{0ex}}\\ \le \underset{j\to \infty }{\text{lim}}\text{sup}{\left(\underset{{B}_{j}\{B}_{r}}{\int }|{u}_{{n}_{j}}{|}^{p}\right)}^{\frac{p-1}{p}}{\left(\underset{{ℝ}^{N}}{\int }|\phi {|}^{p}\right)}^{\frac{1}{p}}\phantom{\rule{2em}{0ex}}\\ \le \underset{j\to \infty }{\text{lim}}\text{sup}{\left(\underset{{B}_{j}\{B}_{r}}{\int }|{u}_{{n}_{j}}{|}^{p}\right)}^{\frac{p-1}{p}}\phantom{\rule{2em}{0ex}}\\ =0\phantom{\rule{2em}{0ex}}\end{array}$

and

$\begin{array}{ll}\hfill \underset{j\to \infty }{\text{lim}}\text{sup}\underset{{B}_{j}\{B}_{r}}{\int }|{u}_{{n}_{j}}{|}^{p-1}|\phi |& \le \phantom{\rule{1em}{0ex}}\underset{j\to \infty }{\text{lim}}\text{sup}{\left(\underset{{B}_{j}\{B}_{r}}{\int }|{u}_{{n}_{j}}{|}^{p}\right)}^{\frac{q-1}{p}}{\left(\underset{{B}_{j}\{B}_{r}}{\int }|\phi {|}^{q}\right)}^{\frac{1}{q}}\phantom{\rule{2em}{0ex}}\\ \le \underset{j\to \infty }{\text{lim}}\text{sup}{\left(\underset{{B}_{j}\{B}_{r}}{\int }|{u}_{{n}_{j}}{|}^{q}\right)}^{\frac{q-1}{q}}{\left(\underset{{ℝ}^{N}}{\int }|\phi {|}^{q}\right)}^{\frac{1}{q}}\phantom{\rule{2em}{0ex}}\\ \le \underset{j\to \infty }{\text{lim}}\text{sup}{\left(\underset{{B}_{j}\{B}_{r}}{\int }|{u}_{{n}_{j}}{|}^{q}\right)}^{\frac{q-1}{q}}\phantom{\rule{2em}{0ex}}\\ =0\phantom{\rule{2em}{0ex}}\end{array}$

Similarly, we get

$\underset{j\to \infty }{\text{lim}}\text{sup}\underset{{B}_{j}\{B}_{r}}{\int }\left(|{v}_{{n}_{j}}{|}^{p-1}|+|{ũ}_{j}{|}^{p-1}+|{ṽ}_{j}{|}^{p-1}\right)\phi =0$

and

$\underset{j\to \infty }{\text{lim}}\text{sup}\underset{{B}_{j}\{B}_{r}}{\int }\left(|{v}_{{n}_{j}}{|}^{q-1}|+|{ũ}_{j}{|}^{q-1}+|{ṽ}_{j}{|}^{q-1}\right)\phi =0.$

Thus

$\underset{j\to \infty }{\text{lim}}\underset{{ℝ}^{N}}{\int }\left({H}_{s}\left({u}_{{n}_{j}},\phantom{\rule{2.77695pt}{0ex}}{v}_{{n}_{j}}\right)-{H}_{s}\left({u}_{{n}_{j}}-{ũ}_{j},\phantom{\rule{2.77695pt}{0ex}}{v}_{{n}_{j}}-{ṽ}_{j}\right)-{H}_{s}\left({ũ}_{j},\phantom{\rule{2.77695pt}{0ex}}{ṽ}_{j}\right)\right)\phi =0.$

From the similar argument, we also get

$\underset{j\to \infty }{\text{lim}}\underset{{ℝ}^{N}}{\int }\left({H}_{t}\left({u}_{{n}_{j}},\phantom{\rule{2.77695pt}{0ex}}{v}_{{n}_{j}}\right)-{H}_{t}\left({u}_{{n}_{j}}-{ũ}_{j},\phantom{\rule{2.77695pt}{0ex}}{v}_{{n}_{j}}-{ṽ}_{j}\right)-{H}_{t}\left({ũ}_{j},\phantom{\rule{2.77695pt}{0ex}}{{ṽ}^{\prime }}_{j}\right)\right)\psi =0.$

Lemma 3.4. One has along a subsequence

${I}_{\lambda }\left({u}_{n}-{ũ}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}-{ṽ}_{n}\right)\to c-{I}_{\lambda }\left(u,\phantom{\rule{2.77695pt}{0ex}}v\right)$

and

Proof. From the Lemma 2.1 of  and the argument of , we have

$\begin{array}{l}\phantom{\rule{1em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{I}_{\lambda }\left({u}_{n}-{ũ}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}-{ṽ}_{n}\right)\phantom{\rule{2em}{0ex}}\\ =\frac{1}{p}\underset{{ℝ}^{N}}{\int }\left(|\nabla {u}_{n}-\nabla {ũ}_{n}{|}^{p}+\lambda V\left(x\right)|{u}_{n}-{ũ}_{n}{|}^{p}+|\nabla {v}_{n}-\nabla {ṽ}_{n}{|}^{p}+\lambda V\left(x\right)|{v}_{n}-{ṽ}_{n}{|}^{p}\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}-\frac{\lambda }{{p}^{*}}\underset{{ℝ}^{N}}{\int }K\left(x\right)\left(|{u}_{n}-{ũ}_{n}{|}^{{p}^{*}}+|{v}_{n}-{ṽ}_{n}{|}^{{p}^{*}}\right)-\lambda \underset{{ℝ}^{N}}{\int }H\left({u}_{n}-{ũ}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}-{ṽ}_{n}\right)\phantom{\rule{2em}{0ex}}\\ ={I}_{\lambda }\left({u}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}\right)-{I}_{\lambda }\left({ũ}_{n},\phantom{\rule{2.77695pt}{0ex}}{ṽ}_{n}\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+\frac{\lambda }{{p}^{*}}\underset{{ℝ}^{N}}{\int }K\left(x\right)\left(\left(|{u}_{n}{|}^{{p}^{*}}-|{u}_{n}-{ũ}_{n}{|}^{{p}^{*}}-|{ũ}_{n}{|}^{{p}^{*}}\right)+\left(|{v}_{n}{|}^{{p}^{*}}-|{v}_{n}-{ṽ}_{n}{|}^{{p}^{*}}-|{ṽ}_{n}{|}^{{p}^{*}}\right)\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+\lambda \underset{{ℝ}^{N}}{\int }\left(H\left({u}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}\right)-H\left({u}_{n}-{ũ}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}-{ṽ}_{n}\right)-H\left({ũ}_{n},\phantom{\rule{2.77695pt}{0ex}}{ṽ}_{n}\right)\right)+o\left(1\right).\phantom{\rule{2em}{0ex}}\end{array}$

By (3.1) and the similar idea of proving the Brézis-Lieb Lemma , 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

$\underset{n\to \infty }{\text{lim}}\underset{{ℝ}^{N}}{\int }\left(H\left({u}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}\right)-H\left({u}_{n}-{ũ}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}{ṽ}_{n}\right)-H\left({ũ}_{n},\phantom{\rule{2.77695pt}{0ex}}{ṽ}_{n}\right)\right)=0.$

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

${I}_{\lambda }\left({u}_{n}-{ũ}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}-{ṽ}_{n}\right)\to c-{I}_{\lambda }\left(u,\phantom{\rule{2.77695pt}{0ex}}v\right).$

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

$\begin{array}{l}\phantom{\rule{0.5em}{0ex}}{I}_{\lambda }^{\text{'}}\left({u}_{n}-{\stackrel{˜}{u}}_{n},{v}_{n}-{\stackrel{˜}{v}}_{n}\right)\left(\phi ,\psi \right)\\ ={I}_{\lambda }^{\text{'}}\left({u}_{n},{v}_{n}\right)\left(\phi ,\psi \right)-{I}_{\lambda }^{\text{'}}\left({\stackrel{˜}{u}}_{n},{\stackrel{˜}{v}}_{n}\right)\left(\phi ,\psi \right)\\ \phantom{\rule{0.5em}{0ex}}+\lambda \underset{{ℝ}^{N}}{\int }K\left(x\right)\left[\left(|{u}_{n}{|}^{{p}^{*}-2}{u}_{n}-|{u}_{n}-{\stackrel{˜}{u}}_{n}{|}^{{p}^{*}-2}\left({u}_{n}-{\stackrel{˜}{u}}_{n}\right)-|{\stackrel{˜}{u}}_{n}{|}^{{p}^{*}-2}{\stackrel{˜}{u}}_{n}\right)\phi \\ \phantom{\rule{0.5em}{0ex}}+\left(|{v}_{n}{|}^{{p}^{*}-2}{v}_{n}-|{v}_{n}-{\stackrel{˜}{v}}_{n}{|}^{{p}^{*}-2}\left({v}_{n}-{\stackrel{˜}{v}}_{n}\right)-|{\stackrel{˜}{v}}_{n}{|}^{{p}^{*}-2}{\stackrel{˜}{v}}_{n}\right)\psi \right]\\ \phantom{\rule{0.5em}{0ex}}+\lambda \underset{{ℝ}^{N}}{\int }\left[\left({H}_{s}\left({u}_{n},\phantom{\rule{0.5em}{0ex}}{v}_{n}\right)-{H}_{s}\left({u}_{n}-{\stackrel{˜}{u}}_{n},\phantom{\rule{0.5em}{0ex}}{v}_{n}-{\stackrel{˜}{v}}_{n}\right)-{H}_{s}\left({\stackrel{˜}{u}}_{n},\phantom{\rule{0.5em}{0ex}}{\stackrel{˜}{v}}_{n}\right)\right)\phi \\ \phantom{\rule{0.5em}{0ex}}+\left({H}_{t}\left({u}_{n},\phantom{\rule{0.5em}{0ex}}{v}_{n}\right)-{H}_{t}\left({u}_{n}-{\stackrel{˜}{u}}_{n},\phantom{\rule{0.5em}{0ex}}{v}_{n}-{\stackrel{˜}{v}}_{n}\right)-{H}_{t}\left({\stackrel{˜}{u}}_{n},\phantom{\rule{0.5em}{0ex}}{\stackrel{˜}{v}}_{n}\right)\right)\psi \right]+o\left(1\right).\end{array}$

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

$\underset{n\to \infty }{\text{lim}}\underset{{ℝ}^{N}}{\int }\left({H}_{s}\left({u}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}\right)-{H}_{s}\left({u}_{n}-{ũ}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}-{ṽ}_{n}\right)-{H}_{s}\left({ũ}_{n},\phantom{\rule{2.77695pt}{0ex}}{ṽ}_{n}\right)\right)\phi =0$

and

$\underset{n\to \infty }{\text{lim}}\underset{{ℝ}^{N}}{\int }\left({H}_{t}\left({u}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}\right)-{H}_{t}\left({u}_{n}-{ũ}_{n},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}-{ṽ}_{n}\right)-{H}_{t}\left({ũ}_{n},\phantom{\rule{2.77695pt}{0ex}}{ṽ}_{n}\right)\right)\psi =0$

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

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

$\begin{array}{l}\phantom{\rule{1em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}{I}_{\lambda }\left({u}_{n}^{1},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}^{1}\right)-\frac{1}{p}{I}_{\lambda }^{\text{'}}\left({u}_{n}^{1},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}^{1}\right)\left({u}_{n}^{1},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}^{1}\right)\phantom{\rule{2em}{0ex}}\\ =\left(\frac{1}{p}-\frac{1}{{p}^{*}}\right)\lambda \underset{{ℝ}^{N}}{\int }K\left(x\right)\left(|{u}_{n}^{1}{|}^{{p}^{*}}+|{v}_{n}^{1}{|}^{{p}^{*}}\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+\lambda \underset{{ℝ}^{N}}{\int }\left(\frac{1}{p}\left({u}_{n}^{1}{H}_{s}\left({u}_{n}^{1},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}^{1}\right)+{v}_{n}^{1}{H}_{t}\left({u}_{n}^{1},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}^{1}\right)\right)-H\left({u}_{n}^{1},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}^{1}\right)\right)\phantom{\rule{2em}{0ex}}\\ \ge \frac{\lambda }{N}\underset{{ℝ}^{N}}{\int }K\left(x\right)\left(|{u}_{n}^{1}{|}^{{p}^{*}}+|{v}_{n}^{1}{|}^{{p}^{*}}\right)\phantom{\rule{2em}{0ex}}\\ \ge \frac{\lambda }{N}{K}_{\text{min}}\underset{{ℝ}^{N}}{\int }\left(|{u}_{n}^{1}{|}^{{p}^{*}}+|{v}_{n}^{1}{|}^{{p}^{*}}\right),\phantom{\rule{2em}{0ex}}\end{array}$

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

Thus by Lemma 3.4, we get

$||\left({u}_{n}^{1},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}^{1}\right)|{|}_{{p}^{*}}^{{p}^{*}}\le \frac{N\left(c-{I}_{\lambda }\left(u,v\right)\right)}{\lambda {K}_{\text{min}}}+o\left(1\right).$
(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

$\begin{array}{l}\underset{{ℝ}^{N}}{\int }\left(K\left(x\right)\left(|u{|}^{{p}^{*}}+|v{|}^{{p}^{*}}\right)+u{H}_{s}\left(u,\phantom{\rule{2.77695pt}{0ex}}v\right)+v{H}_{t}\left(u,\phantom{\rule{2.77695pt}{0ex}}v\right)\right)\phantom{\rule{2em}{0ex}}\\ \hfill \le & b\left(||u|{|}_{p}^{p}+||v|{|}_{p}^{p}\right)+{C}_{b}\left(||u|{|}_{{p}^{*}}^{{p}^{*}}+||v|{|}_{{p}^{*}}^{{p}^{*}}\right).\phantom{\rule{2em}{0ex}}\end{array}$
(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 {(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

$\mathrm{lim}\underset{n\to \infty }{\mathrm{inf}}||\left({u}_{n}^{1},\phantom{\rule{0.5em}{0ex}}{v}_{n}^{1}\right)|{|}_{\lambda }>0$

and

$c-{J}_{\lambda }\left(u,\phantom{\rule{2.77695pt}{0ex}}v\right)>0.$

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

$\begin{array}{l}\phantom{\rule{1em}{0ex}}S\left(||{u}_{n}^{1}|{|}_{{p}^{*}}^{p}+||{v}_{n}^{1}|{|}_{{p}^{*}}^{p}\right)\phantom{\rule{2em}{0ex}}\\ \le \underset{{ℝ}^{N}}{\int }\left(|\nabla {u}_{n}^{1}{|}^{p}+|\nabla {v}_{n}^{1}{|}^{p}\right)\phantom{\rule{2em}{0ex}}\\ =\underset{{ℝ}^{N}}{\int }\left(|\nabla {u}_{n}^{1}{|}^{p}+\lambda V\left(x\right)|{u}_{n}^{1}{|}^{p}+|\nabla {v}_{n}^{1}{|}^{p}+\lambda V\left(x\right)|{v}_{n}^{1}{|}^{p}\right)-\lambda \underset{{ℝ}^{N}}{\int }V\left(x\right)\left(|{u}_{n}^{1}{|}^{p}+|{v}_{n}^{1}{|}^{p}\right)\phantom{\rule{2em}{0ex}}\\ =\lambda \underset{{ℝ}^{N}}{\int }K\left(x\right)\left(|{u}_{n}^{1}{|}^{{p}^{*}}+|{v}_{n}^{1}{|}^{{p}^{*}}\right)+{u}_{n}^{1}{H}_{s}\left({u}_{n}^{1},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}^{1}\right)+{v}_{n}^{1}{H}_{t}\left({u}_{n}^{1},\phantom{\rule{2.77695pt}{0ex}}{v}_{n}^{1}\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}-\lambda \underset{{ℝ}^{N}}{\int }V\left(x\right)\left(|{u}_{n}^{1}{|}^{p}+|{v}_{n}^{1}{|}^{p}\right)+o\left(1\right)\phantom{\rule{2em}{0ex}}\\ \le \lambda b\left(||{u}_{n}^{1}|{|}_{p}^{p}+||{v}_{n}^{1}|{|}_{p}^{p}\right)+\lambda {C}_{b}\left(||{u}_{n}^{1}|{|}_{{p}^{*}}^{{p}^{*}}+||{v}_{n}^{1}|{|}_{{p}^{*}}^{{p}^{*}}\right)-\lambda b\left(||{u}_{n}^{1}|{|}_{p}^{p}+||{v}_{n}^{1}|{|}_{p}^{p}\right)+o\left(1\right)\phantom{\rule{2em}{0ex}}\\ =\lambda {C}_{b}\left(||{u}_{n}^{1}|{|}_{{p}^{*}}^{{p}^{*}}+||{v}_{n}^{1}|{|}_{{p}^{*}}^{{p}^{*}}\right)+o\left(1\right).\phantom{\rule{2em}{0ex}}\end{array}$

This, together with $\mathrm{lim}\phantom{\rule{0.5em}{0ex}}{\mathrm{inf}}_{n\to \infty }\left(||{u}_{n}^{1}|{|}_{{p}^{*}}^{{p}^{*}}+||{v}_{n}^{1}|{|}_{{p}^{*}}^{{p}^{*}}\right)>0$ and (3.3), gives

$\begin{array}{ll}\hfill S& \le \lambda {C}_{b}{\left(||{u}_{n}^{1}|{|}_{{p}^{*}}^{{p}^{*}}+||{v}_{n}^{1}|{|}_{{p}^{*}}^{{p}^{*}}\right)}^{\frac{{p}^{*}-p}{{p}^{*}}}+o\left(1\right)\phantom{\rule{2em}{0ex}}\\ \le \lambda {C}_{b}{\left(\frac{N\left(c-{I}_{\lambda }\left(u,v\right)\right)}{\lambda {K}_{\text{min}}}\right)}^{\frac{p}{N}}+o\left(1\right)\phantom{\rule{2em}{0ex}}\\ ={\lambda }^{1-\frac{p}{N}}{C}_{b}{\left(\frac{N}{{K}_{\text{min}}}\right)}^{\frac{p}{N}}{\left(c-{I}_{\lambda }\left(u,\phantom{\rule{2.77695pt}{0ex}}v\right)\right)}^{\frac{p}{N}}+o\left(1\right).\phantom{\rule{2em}{0ex}}\end{array}$

Set ${\alpha }_{0}={S}^{\frac{N}{p}}{C}_{b}^{-\frac{N}{p}}{N}^{-1}{K}_{\text{min}}$, then

${\alpha }_{0}{\lambda }^{1-\frac{N}{p}}\le c-{I}_{\lambda }\left(u,\phantom{\rule{2.77695pt}{0ex}}v\right)+o\left(1\right).$

This proof is completed. □

Since ${W}^{1,p}\left({ℝ}^{N}\right)↪{L}^{{p}^{*}}\left({ℝ}^{N}\right)$ 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

$c-{I}_{\lambda }\left(u,\phantom{\rule{2.77695pt}{0ex}}v\right)\le {\alpha }_{0}{\lambda }^{1-\frac{N}{p}}-{I}_{\lambda }\left(u,\phantom{\rule{2.77695pt}{0ex}}v\right).$

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

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

$\underset{{ℝ}^{N}}{\int }G\left(u,\phantom{\rule{2.77695pt}{0ex}}v\right)\le \delta \left(||u|{|}_{p}^{p}+||v|{|}_{p}^{p}\right)+{C}_{\delta }\left(||u|{|}_{{p}^{*}}^{{p}^{*}}+||v|{|}_{{p}^{*}}^{{p}^{*}}\right).$

Thus

$\begin{array}{ll}\hfill {I}_{\lambda }\left(u,\phantom{\rule{2.77695pt}{0ex}}v\right)& =\frac{1}{p}||\left(u,\phantom{\rule{2.77695pt}{0ex}}v\right)|{|}_{\lambda }^{p}-\lambda \underset{{ℝ}^{N}}{\int }G\left(u,\phantom{\rule{2.77695pt}{0ex}}v\right)\phantom{\rule{2em}{0ex}}\\ \ge \frac{1}{p}||\left(u,\phantom{\rule{2.77695pt}{0ex}}v\right)|{|}_{\lambda }^{p}-\lambda \delta \left(||u|{|}_{p}^{p}+||v|{|}_{p}^{p}\right)-\lambda {C}_{\delta }\left(||u|{|}_{{p}^{*}}^{{p}^{*}}+||v|{|}_{{p}^{*}}^{{p}^{*}}\right).\phantom{\rule{2em}{0ex}}\end{array}$

Note that $||u|{|}_{p}^{p}+||v|{|}_{p}^{p}\le {C}_{1}||\left(u,\phantom{\rule{2.77695pt}{0ex}}v\right)|{|}_{\lambda }^{p}$. If δ ≤ (2pλC1)-1, then

${I}_{\lambda }\left(u,\phantom{\rule{2.77695pt}{0ex}}v\right)\ge \frac{1}{2p}||\left(u,\phantom{\rule{2.77695pt}{0ex}}v\right)|{|}_{\lambda }^{p}-\lambda {C}_{\delta }\left(||u|{|}_{{p}^{*}}^{{p}^{*}}+||v|{|}_{{p}^{*}}^{{p}^{*}}\right).$

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

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 c λ small sufficiently, I λ satisfies (PS) condition.

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

Define the functional

${\Phi }_{\lambda }\left(u,\phantom{\rule{2.77695pt}{0ex}}v\right)=\frac{1}{p}\underset{{ℝ}^{N}}{\int }\left(|\nabla u{|}^{p}+\lambda V\left(x\right)|u{|}^{p}+|\nabla v{|}^{p}+\lambda V\left(x\right)|v{|}^{p}\right)-\lambda {a}_{0}\underset{{ℝ}^{N}}{\int }\left(|u{|}^{\alpha }+|v{|}^{\beta }\right).$

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

Observe that

$\text{inf}\left\{\underset{{ℝ}^{N}}{\int }|\nabla \varphi {|}^{p}:\varphi \in {C}_{0}^{\infty }\left({ℝ}^{N},ℝ\right),\phantom{\rule{2.77695pt}{0ex}}|\varphi {|}_{{L}^{\alpha }\left({ℝ}^{N}\right)}=1\right\}=0$

and

$\text{inf}\left\{\underset{{ℝ}^{N}}{\int }|\nabla \psi {|}^{p}:\psi \in {C}_{0}^{\infty }\left({ℝ}^{N},ℝ\right),\phantom{\rule{2.77695pt}{0ex}}|\psi {|}_{{L}^{\beta }\left({ℝ}^{N}\right)}=1\right\}=0.$

For any δ> 0, there are φ δ , ${\psi }_{\delta }\in {C}_{0}^{\infty }\left({ℝ}^{N},ℝ\right)$ with $|{\varphi }_{\delta }{|}_{{L}^{\alpha }\left({ℝ}^{N}\right)}=\phantom{\rule{2.77695pt}{0ex}}|{\psi }_{\delta }{|}_{{L}^{\beta }\left({ℝ}^{N}\right)}=1$ and suppφδ, $\text{supp}{\psi }_{\delta }\subset {B}_{{r}_{\delta }}\left(0\right)$ such that $|\nabla {\varphi }_{\delta }{|}_{p}^{p},|\nabla {\psi }_{\delta }{|}_{p}^{p}<\delta$.

Let ${w}_{\lambda }\left(x\right)=\left({\varphi }_{\delta }\left(\sqrt[p]{\lambda }x\right),\phantom{\rule{2.77695pt}{0ex}}{\psi }_{\delta }\left(\sqrt[p]{\lambda }x\right)\right)$, then $\text{supp}{w}_{\lambda }\subset {B}_{{\lambda }^{-\frac{1}{p}{r}_{\phantom{\rule{0.3em}{0ex}}\delta }}}\left(0\right)$. For t ≥ 0, we get

$\begin{array}{ll}\hfill {\Phi }_{\lambda }\left(t{w}_{\lambda }\right)& =\frac{{t}^{p}}{p}{∥{w}_{\lambda }∥}_{\lambda }^{p}-{a}_{0}\lambda {t}^{\alpha }{\int }_{{ℝ}^{N}}|{\varphi }_{\delta }\left(\sqrt[p]{\lambda }x\right){|}^{\alpha }-{a}_{0}\lambda {t}^{\beta }\underset{{ℝ}^{N}}{\int }|{\psi }_{\delta }\left(\sqrt[p]{\lambda }x\right){|}^{\beta }\phantom{\rule{2em}{0ex}}\\ ={\lambda }^{1-\frac{N}{p}}{J}_{\lambda }\left(t{\varphi }_{\delta },t{\psi }_{\delta }\right),\phantom{\rule{2em}{0ex}}\end{array}$

where

${J}_{\lambda }\left(u,\phantom{\rule{2.77695pt}{0ex}}v\right)=\frac{1}{p}\underset{{ℝ}^{N}}{\int }\left(|\nabla u{|}^{p}+|\nabla v{|}^{p}+V\left({\lambda }^{-\frac{1}{p}}x\right)\left(|u{|}^{p}+|v{|}^{p}\right)\right)-{a}_{0}\underset{{ℝ}^{N}}{\int }\left(|u{|}^{\alpha }+|v{|}^{\beta }\right).$

We easily prove that

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

Together with V (0) = 0 and $|\nabla {\varphi }_{\delta }{|}_{p}^{p},\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}|\nabla {\psi }_{\delta }{|}_{p}^{p}<\delta$, this implies that there is Λδ> 0 such that for all λ ≥ Λδ, we have

$\underset{t\ge 0}{\text{max}}{I}_{\lambda }\left(t{\varphi }_{\delta },\phantom{\rule{2.77695pt}{0ex}}t{\psi }_{\delta }\right)\le \left(\frac{\alpha -p}{p\alpha {\left(\alpha {a}_{0}\right)}^{\frac{p}{\alpha -p}}}{\left(2\delta \right)}^{\frac{\alpha }{\alpha -p}}+\frac{\beta -p}{p\beta {\left(\beta {a}_{0}\right)}^{\frac{p}{\beta -p}}}{\left(2\delta \right)}^{\frac{\beta }{\beta -p}}\right){\lambda }^{1-\frac{N}{p}}.$
(3.6)

It follows from (3.6) that

Lemma 3.9. Under the assumptions of Lemma 3.7, for any > 0, there is Λσ> 0 such that λ ≥ Λσ, there exists ${\stackrel{̄}{w}}_{\lambda }\in B$ with ${∥{\stackrel{̄}{w}}_{\lambda }∥}_{\lambda }>{\rho }_{\lambda },{I}_{\lambda }\left({\stackrel{̄}{w}}_{\lambda }\right)\le 0$ and

$\underset{t\ge 0}{\text{max}}{I}_{\lambda }\left(t{\stackrel{̄}{w}}_{\lambda }\right)\le \sigma {\lambda }^{1\phantom{\rule{0.3em}{0ex}}-\phantom{\rule{0.3em}{0ex}}\frac{N}{p}},$

where ρ λ is defined in Lemma 3.7.

Proof. This proof is similar to the one of Lemma 4.3 in , 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 ${\Gamma }_{\lambda }=\left\{\gamma \in C\left(\left[0,1\right],\phantom{\rule{2.77695pt}{0ex}}B\right):\gamma \left(0\right)=0,\phantom{\rule{2.77695pt}{0ex}}\gamma \left(1\right)={\stackrel{̄}{w}}_{\lambda }\right\}.$

Furthermore, Lemma 3.6 implies that I λ satisfies (PS) condition. Hence, by the mountain-pass theorem, there is (u λ , v λ ) B satisfying I λ (u λ , v λ ) = c λ and ${I}_{\lambda }^{\prime }\left({u}_{\lambda },\phantom{\rule{2.77695pt}{0ex}}{v}_{\lambda }\right)=0.$ This shows (u λ , v λ ) is a weak solution of (2.1). Similar to the argument in , 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 ${I}_{\lambda }\left({u}_{\lambda },\phantom{\rule{2.77695pt}{0ex}}{v}_{\lambda }\right)\le \sigma {\lambda }^{1-\frac{N}{p}}$ and ${I}_{\lambda }^{\prime }\left({u}_{\lambda },\phantom{\rule{2.77695pt}{0ex}}{v}_{\lambda }\right)=0.$ we have

$\begin{array}{ll}\hfill {I}_{\lambda }\left({u}_{\lambda },{v}_{\lambda }\right)& ={I}_{\lambda }\left({u}_{\lambda },{v}_{\lambda }\right)-\frac{1}{\theta }{I}_{\lambda }^{\text{'}}\left({u}_{\lambda },{v}_{\lambda }\right)\left({u}_{\lambda },{v}_{\lambda }\right)\phantom{\rule{2em}{0ex}}\\ =\left(\frac{1}{p}-\frac{1}{\theta }\right){∥\left({u}_{\lambda },{v}_{\lambda }\right)∥}_{\lambda }^{p}+\left(\frac{1}{\theta }-\frac{1}{{p}^{*}}\right)\lambda \underset{{ℝ}^{N}}{\int }K\left(x\right)\left(|{u}_{\lambda }{|}^{{p}^{*}}+|{v}_{\lambda }{|}^{{p}^{*}}\right)\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}+\lambda \underset{{ℝ}^{N}}{\int }\left(\frac{1}{\theta }\left({u}_{\lambda }{H}_{s}\left({u}_{\lambda },{v}_{\lambda }\right)+{v}_{\lambda }{H}_{t}\left({u}_{\lambda },{v}_{\lambda }\right)\right)-H\left({u}_{\lambda },{v}_{\lambda }\right)\right)\phantom{\rule{2em}{0ex}}\\ \ge \left(\frac{1}{p}-\frac{1}{\theta }\right){∥\left({u}_{\lambda },{v}_{\lambda }\right)∥}_{\lambda }^{p}.\phantom{\rule{2em}{0ex}}\end{array}$

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

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## 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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Correspondence to Huixing Zhang.

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Zhang, H., Liu, W. Existence of nontrivial solutions to perturbed p-Laplacian system in N involving critical nonlinearity. Bound Value Probl 2012, 53 (2012). https://doi.org/10.1186/1687-2770-2012-53 