Existence of solutions for a general quasilinear elliptic system via perturbation method

Yujuan Jiao; Shengmao Fu; Yanli Wang

doi:10.1186/1687-2770-2013-219

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Existence of solutions for a general quasilinear elliptic system via perturbation method

Yujuan Jiao¹Email author,
Shengmao Fu² and
Yanli Wang³

Boundary Value Problems20132013:219

https://doi.org/10.1186/1687-2770-2013-219

Received: 28 May 2013

Accepted: 28 August 2013

Published: 7 November 2013

Abstract

In this paper, we consider the following quasilinear elliptic system:

{\begin{matrix} - \sum_{i, j = 1}^{N} D_{j} (a_{i j} (x, u) D_{i} u) + \frac{1}{2} \sum_{i, j = 1}^{N} D_{s} a_{i j} (x, u) D_{i} u D_{j} u = \frac{2 α}{α + β} {| u |}^{α - 2} {| v |}^{β} u, & x \in Ω, \\ - \sum_{i, j = 1}^{N} D_{j} (b_{i j} (x, v) D_{i} v) + \frac{1}{2} \sum_{i, j = 1}^{N} D_{s} b_{i j} (x, v) D_{i} v D_{j} v = \frac{2 β}{α + β} {| u |}^{α} {| v |}^{β - 2} v, & x \in Ω, \\ u = 0, v = 0, & x \in \partial Ω, \end{matrix}

where $D_{i} u = \frac{\partial u}{\partial x_{i}}$ , $D_{s} a_{i j} (x, u) = \frac{\partial}{\partial u} a_{i j} (x, u)$ , $D_{s} b_{i j} (x, v) = \frac{\partial}{\partial v} b_{i j} (x, v)$ , $α > 2$ , $β > 2$ , $α + β < 2 \cdot 2^{*}$ , $2^{*} = \frac{2 N}{N - 2}$ is the critical Sobolev exponent and $Ω \subset R^{N}$ ( $N \geq 3$ ) is a bounded smooth domain. By using the perturbation method, we establish the existence of both positive and negative solutions for this system.

MSC:35J60, 35B33.

Keywords

quasilinear elliptic system positive solution negative solution perturbation method

1 Introduction

Let us consider the following quasilinear elliptic system:

{\begin{matrix} - \sum_{i, j = 1}^{N} D_{j} (a_{i j} (x, u) D_{i} u) + \frac{1}{2} \sum_{i, j = 1}^{N} D_{s} a_{i j} (x, u) D_{i} u D_{j} u = \frac{2 α}{α + β} {| u |}^{α - 2} {| v |}^{β} u, & x \in Ω, \\ - \sum_{i, j = 1}^{N} D_{j} (b_{i j} (x, v) D_{i} v) + \frac{1}{2} \sum_{i, j = 1}^{N} D_{s} b_{i j} (x, v) D_{i} v D_{j} v = \frac{2 β}{α + β} {| u |}^{α} {| v |}^{β - 2} v, & x \in Ω, \\ u = 0, v = 0, & x \in \partial Ω, \end{matrix}

(1.1)

where

D_{i} u = \frac{\partial u}{\partial x_{i}}

,

D_{s} a_{i j} (x, u) = \frac{\partial}{\partial u} a_{i j} (x, u)

,

D_{s} b_{i j} (x, v) = \frac{\partial}{\partial v} b_{i j} (x, v)

,

α > 2

,

β > 2

,

α + β < 2 \cdot 2^{*}

,

2^{*} = \frac{2 N}{N - 2}

is the critical Sobolev exponent and

Ω \subset R^{N}

(

N \geq 3

) is a bounded smooth domain. This system includes the following special class of system with

a_{i j} (x, u) = (1 + u^{2}) δ_{i j}

,

b_{i j} (x, v) = (1 + v^{2}) δ_{i j}

, i.e.,

{\begin{matrix} - △ u - \frac{1}{2} u △ (u^{2}) = \frac{2 α}{α + β} {| u |}^{α - 2} {| v |}^{β} u, & x \in Ω, \\ - △ v - \frac{1}{2} v △ (v^{2}) = \frac{2 β}{α + β} {| u |}^{α} {| v |}^{β - 2} v, & x \in Ω, \\ u = 0, v = 0, & x \in \partial Ω, \end{matrix}

which is referred to as the so-called modified nonlinear Schrödinger system.

Our assumptions on the functions $a_{i j}$ and $b_{i j}$ are as follows.

(A₁) The functions $a_{i j} \in C^{1} (\bar{Ω} \times R, R)$ , $b_{i j} \in C^{1} (\bar{Ω} \times R, R)$ , $a_{i j} = a_{j i}$ , $b_{i j} = b_{j i}$ , $i, j = 1, 2, \dots, N$ .
(A₂) There exist constants $a_{0}$ , $a_{1}$ , $b_{0}$ , $b_{1}$ satisfying $a_{1} \geq a_{0} > 0$ , $b_{1} \geq b_{0} > 0$ , $(α + β - 2) a_{0} > 2 a_{1}$ and $(α + β - 2) b_{0} > 2 b_{1}$ such that
$\begin{matrix} a_{0} (1 + s^{2}) {| ξ |}^{2} \leq \sum_{i, j = 1}^{N} a_{i j} (x, s) ξ_{i} ξ_{j} \leq a_{1} (1 + s^{2}) {| ξ |}^{2}, \\ b_{0} (1 + s^{2}) {| ξ |}^{2} \leq \sum_{i, j = 1}^{N} b_{i j} (x, s) ξ_{i} ξ_{j} \leq b_{1} (1 + s^{2}) {| ξ |}^{2} \end{matrix}$

for $x \in \bar{Ω}$ , $ξ \in R^{N}$ , $s \in R$ .
(A₃)
$\begin{matrix} 0 \leq \sum_{i, j = 1}^{N} D_{s} a_{i j} (x, s) s ξ_{i} ξ_{j} \leq 2 \sum_{i, j = 1}^{N} a_{i j} (x, s) ξ_{i} ξ_{j}, \\ 0 \leq \sum_{i, j = 1}^{N} D_{s} b_{i j} (x, s) s ξ_{i} ξ_{j} \leq 2 \sum_{i, j = 1}^{N} b_{i j} (x, s) ξ_{i} ξ_{j} \end{matrix}$

for $x \in \bar{Ω}$ , $ξ \in R^{N}$ , $s \in R$ .

In recent years, much attention has been devoted to the quasilinear Schrödinger equation of the following form:

- △ u + λ V (x) u - k △ (u^{2}) u = {| u |}^{p - 2} u, x \in R^{N} .

(1.2)

See, for example, [1] where Poppenberg et al. proved the existence of a positive ground state solution by using a constrained minimization argument. Using a change of variables, Liu et al. [2] used an Orlicz space to prove the existence of a soliton solution for equation (1.2) via the mountain pass theorem. Colin and Jeanjean [3] also made use of a change of variables but worked in the Sobolev space

H^{1} (R^{N})

. They proved the existence of a positive solution for equation (1.2) from the classical results given by Berestycki and Lions [4]. Liu et al. [5] established the existence of both one-sign and nodal ground states of soliton-type solutions for equation (1.2) by the Nehari method. By using the Nehari manifold method and the concentration compactness principle (see [6]) in the Orlicz space, Guo and Tang [7] considered the following quasilinear Schrödinger system:

{\begin{matrix} - △ u + (λ a (x) + 1) u - \frac{1}{2} (△ {| u |}^{2}) u = \frac{2 α}{α + β} {| u |}^{α - 2} {| v |}^{β} u, & x \in R^{N}, \\ - △ u + (λ b (x) + 1) u - \frac{1}{2} (△ {| u |}^{2}) u = \frac{2 β}{α + β} {| u |}^{α} {| v |}^{β - 2} v, & x \in R^{N}, \\ u (x) \to 0, v (x) \to 0, & | x | \to \infty, \end{matrix}

(1.3)

with

a (x) \geq 0

,

b (x) \geq 0

having a potential well and

α > 2

,

β > 2

,

α + β < 2 \cdot 2^{*}

, and they proved the existence of a ground state solution for system (1.3) which localizes near the potential well

int a^{- 1} (0)

for λ large enough. Guo and Tang [8] considered also ground state solutions of the single quasilinear Schrödinger equation corresponding to system (1.3) by the same methods and obtained similar results. In particular, by the perturbation method, Liu et al. [9] considered the existence and multiplicity of solutions for the following quasilinear equation of the form

{\begin{matrix} \sum_{i, j = 1}^{N} D_{j} (a_{i j} (x, u) D_{i} u) - \frac{1}{2} \sum_{i, j = 1}^{N} D_{s} a_{i j} (x, u) D_{i} u D_{j} u + f (x, u) = 0, & x \in Ω, \\ u = 0, & x \in \partial Ω \end{matrix}

(1.4)

under suitable assumptions.

It is worth pointing out that the existence of one-bump or multi-bump bound state solutions for the related semilinear Schrödinger equation (1.2) for $k = 0$ has been extensively studied. One can see Bartsch and Wang [10], Ambrosetti et al. [11], Ambrosetti et al. [12], Byeon and Wang [13], Cingolani and Lazzo [14], Cingolani and Nolasco [15], Del Pino and Felmer [16, 17], Floer and Weinstein [18], Oh [19, 20] and the references therein.

Motivated by the single equation (1.4), the purpose of this paper is to study the existence of both positive and negative solutions for the coupled quasilinear system (1.1). We mainly follow the idea of Liu et al. [9] to perturb the functional and obtain our main results. We point out that the procedure to system (1.1) is not trivial at all. Since the appearance of the quasilinear terms $\sum_{i, j = 1}^{N} D_{j} (a_{i j} (x, u) D_{i} u) - \frac{1}{2} \sum_{i, j = 1}^{N} D_{s} a_{i j} (x, u) D_{i} u D_{j} u$ and $\sum_{i, j = 1}^{N} D_{j} (b_{i j} (x, v) D_{i} v) - \frac{1}{2} \sum_{i, j = 1}^{N} D_{s} b_{i j} (x, v) D_{i} v D_{j} v$ , we need more delicate estimates.

The paper is organized as follows. In Section 2, we introduce a perturbation of the functional and give our main results (Theorem 2.1 and Theorem 2.2). In Section 3, we verify the Palais-Smale condition for the perturbed functional. Section 4 is devoted to some asymptotic behavior of the sequences ${(u_{n}, v_{n})} \subset W_{0}^{1, 4} (Ω) \times W_{0}^{1, 4} (Ω)$ and ${μ_{n}} \subset (0, 1]$ satisfying some conditions. Finally, our main results will be proved in Section 5.

Throughout this paper, we will use the same C to denote various generic positive constants, and we will use $o (1)$ to denote quantities that tend to 0.

2 Perturbation of the functional and main results

In order to obtain the desired existence of solutions for system (1.1), in this section, we introduce a perturbation of the functional and give our main results.

The weak form of system (1.1) is

\begin{aligned} \int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, u) D_{i} u D_{j} φ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} D_{s} a_{i j} (x, u) D_{i} u D_{j} u φ \\ + \int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, v) D_{i} v D_{j} ψ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} D_{s} b_{i j} (x, v) D_{i} v D_{j} v ψ \\ - \frac{2 α}{α + β} \int_{Ω} {| u |}^{α - 2} {| v |}^{β} u φ - \frac{2 β}{α + β} \int_{Ω} {| u |}^{α} {| v |}^{β - 2} v ψ = 0 \end{aligned}

(2.1)

for all

(φ, ψ) \in C_{0}^{\infty} (Ω) \times C_{0}^{\infty} (Ω)

, which is formally the variational formulation of the following functional:

I_{0} (u, v) = \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, u) D_{i} u D_{j} u + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, v) D_{i} v D_{j} v - \frac{2}{α + β} \int_{Ω} {| u |}^{α} {| v |}^{β} .

(2.2)

We may define the derivative of

I_{0}

at

(u, v)

in the direction of

(φ, ψ) \in C_{0}^{\infty} (Ω) \times C_{0}^{\infty} (Ω)

as follows:

\begin{array}{rcl} 〈 I_{0}^{'} (u, v), (φ, ψ) 〉 & = & \int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, u) D_{i} u D_{j} φ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} D_{s} a_{i j} (x, u) D_{i} u D_{j} u φ \\ + \int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, v) D_{i} v D_{j} ψ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} D_{s} b_{i j} (x, v) D_{i} v D_{j} v ψ \\ - \frac{2 α}{α + β} \int_{Ω} {| u |}^{α - 2} {| v |}^{β} u φ - \frac{2 β}{α + β} \int_{Ω} {| u |}^{α} {| v |}^{β - 2} v ψ . \end{array}

(2.3)

We call $(u, v)$ a critical point of $I_{0}$ if $(u, v) \in W_{0}^{1, 2} (Ω) \times W_{0}^{1, 2} (Ω)$ , $\int_{Ω} u^{2} {| \nabla u |}^{2} < \infty$ , $\int_{Ω} v^{2} {| \nabla v |}^{2} < \infty$ and $〈 I_{0}^{'} (u, v), (φ, ψ) 〉 = 0$ for all $(φ, ψ) \in C_{0}^{\infty} (Ω) \times C_{0}^{\infty} (Ω)$ . That is, $(u, v)$ is a weak solution for system (1.1).

When we consider system (1.1) by using the classical critical point theory, we encounter the difficulties due to the lack of an appropriate working space. In general, it seems that there is no suitable space in which the variational functional

I_{0}

possesses both smoothness and compactness properties. For smoothness, one would need to work in a space smaller than

W_{0}^{1, 2} (Ω)

to control the term involving the quasilinear term in system (1.1), but it seems impossible to obtain bounds for

{(PS)}_{c}

sequence in this setting. Several ideas and approaches, such as minimizations [1, 21], the Nehari method [5] and change of variables [2, 3], have been used in recent years to overcome the difficulties. In this paper, we consider the perturbed functional

\begin{array}{rcl} I_{μ} (u, v) & = & \frac{1}{4} μ \int_{Ω} ({| \nabla u |}^{4} + {| \nabla v |}^{4}) + I_{0} (u, v) \\ = & \frac{1}{4} μ \int_{Ω} ({| \nabla u |}^{4} + {| \nabla v |}^{4}) + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, u) D_{i} u D_{j} u \\ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, v) D_{i} v D_{j} v - \frac{2}{α + β} \int_{Ω} {| u |}^{α} {| v |}^{β}, \end{array}

(2.4)

where

μ \in (0, 1]

is a parameter. Then it is easy to see that

I_{μ}

is a

C^{1}

-functional on

W_{0}^{1, 4} (Ω) \times W_{0}^{1, 4} (Ω)

. We can define also the derivative of

I_{μ}

at

(u, v)

in the direction of

(φ, ψ)

as follows:

\begin{array}{rcl} 〈 I_{μ}^{'} (u, v), (φ, ψ) 〉 & = & μ \int_{Ω} {| \nabla u |}^{2} \nabla u \nabla φ + μ \int_{Ω} {| \nabla v |}^{2} \nabla v \nabla ψ \\ + \int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, u) D_{i} u D_{j} φ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} D_{s} a_{i j} (x, u) D_{i} u D_{j} u φ \\ + \int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, v) D_{i} v D_{j} ψ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} D_{s} b_{i j} (x, v) D_{i} v D_{j} v ψ \\ - \frac{2 α}{α + β} \int_{Ω} {| u |}^{α - 2} {| v |}^{β} u φ - \frac{2 β}{α + β} \int_{Ω} {| u |}^{α} {| v |}^{β - 2} v ψ \end{array}

(2.5)

for all $(φ, ψ) \in C_{0}^{\infty} (Ω) \times C_{0}^{\infty} (Ω)$ . The idea of this paper is to obtain the existence of the critical points of $I_{μ}$ for $μ > 0$ small and establish suitable estimates for the critical points as $μ \to 0$ so that we may pass to the limit to get the solutions for the original system (1.1).

Our main results are as follows.

Theorem 2.1 Assume that (A₁)-(A₃) hold,

α > 2

,

β > 2

and

α + β < 2 \cdot 2^{*}

. Let

μ_{n} \to 0

and let

{(u_{n}, v_{n})} \subset W_{0}^{1, 4} (Ω) \times W_{0}^{1, 4} (Ω)

be a sequence of critical points of

I_{μ_{n}}

satisfying

I_{μ_{n}}^{'} (u_{n}, v_{n}) = 0

and

I_{μ_{n}} (u_{n}, v_{n}) \leq C

for some C independent of n. Then, up to a subsequence,

\begin{matrix} u_{n} \to u, v_{n} \to v in W_{0}^{1, 2} (Ω), \\ u_{n} \nabla u_{n} \to u \nabla u, v_{n} \nabla v_{n} \to v \nabla v in L^{2} (Ω), \\ μ_{n} \int_{Ω} ({| \nabla u_{n} |}^{4} + {| \nabla v_{n} |}^{4}) \to 0, \\ I_{μ_{n}}^{'} (u_{n}, v_{n}) \to I_{0}^{'} (u, v) \end{matrix}

as $n \to \infty$ , and $(u, v)$ is a critical point of $I_{0}$ .

Theorem 2.2 Assume that (A₁)-(A₃) hold, $α > 2$ , $β > 2$ and $α + β < 2 \cdot 2^{*}$ . Then $I_{μ}$ has a positive critical point $(u_{μ}, v_{μ})$ and a negative critical point $({\tilde{u}}_{μ}, {\tilde{v}}_{μ})$ , and $(u_{μ}, v_{μ})$ (resp., $({\tilde{u}}_{μ}, {\tilde{v}}_{μ})$ ) converges to a positive (resp., negative) solution for system (1.1) as $μ \to 0$ .

Notation We denote by $∥ \cdot ∥$ the norm of $W_{0}^{1, 4} (Ω)$ and by ${| \cdot |}_{s}$ the norm of $L^{s} (Ω)$ ( $1 \leq s < + \infty$ ).

3 Compactness of the perturbed functional

In this section, we verify the Palais-Smale condition ( ${(PS)}_{c}$ condition in short) for the perturbed functional $I_{μ} (u, v)$ . We have the following proposition.

Proposition 3.1 For

μ > 0

fixed, the functional

I_{μ} (u, v)

satisfies

{(PS)}_{c}

condition for all

c \in R

. That is, any sequence

{(u_{n}, v_{n})} \subset W_{0}^{1, 4} (Ω) \times W_{0}^{1, 4} (Ω)

satisfying, for

c \in R

,

I_{μ} (u_{n}, v_{n}) \to c,

(3.1)

I_{μ}^{'} (u_{n}, v_{n}) \to 0 strongly in {(W_{0}^{1, 4} (Ω) \times W_{0}^{1, 4} (Ω))}^{*}

(3.2)

has a strongly convergent subsequence in $W_{0}^{1, 4} (Ω) \times W_{0}^{1, 4} (Ω)$ , where ${(W_{0}^{1, 4} (Ω) \times W_{0}^{1, 4} (Ω))}^{*}$ is the dual space of $W_{0}^{1, 4} (Ω) \times W_{0}^{1, 4} (Ω)$ .

To give the proof of Proposition 3.1, we need the following lemma firstly.

Lemma 3.2 Suppose that a sequence

{(u_{n}, v_{n})} \subset W_{0}^{1, 4} (Ω) \times W_{0}^{1, 4} (Ω)

satisfies (3.1) and (3.2). Then

\underset{n \to \infty}{lim sup} {∥ (u_{n}, v_{n}) ∥}^{4} \leq {(\frac{1}{4} - \frac{1}{α + β})}^{- 1} μ^{- 1} c .

Proof It follows from (3.1) and (3.2) that

\begin{aligned} c + o (1) - \frac{1}{α + β} o (1) ∥ (u_{n}, v_{n}) ∥ \\ = I_{μ} (u_{n}, v_{n}) - \frac{1}{α + β} 〈 I_{μ}^{'} (u_{n}, v_{n}), (u_{n}, v_{n}) 〉 \\ = (\frac{1}{4} - \frac{1}{α + β}) μ \int_{Ω} ({| \nabla u_{n} |}^{4} + {| \nabla v_{n} |}^{4}) \\ + (\frac{1}{2} - \frac{1}{α + β}) \int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, u) D_{i} u D_{j} u + (\frac{1}{2} - \frac{1}{α + β}) \int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, v) D_{i} v D_{j} v \\ - \frac{1}{2 (α + β)} \int_{Ω} \sum_{i, j = 1}^{N} D_{s} a_{i j} (x, u) D_{i} u D_{j} u - \frac{1}{2 (α + β)} \int_{Ω} \sum_{i, j = 1}^{N} D_{s} b_{i j} (x, v) D_{i} v D_{j} v \\ \geq (\frac{1}{4} - \frac{1}{α + β}) μ \int_{Ω} ({| \nabla u |}^{4} + {| \nabla v |}^{4}) + \frac{(α + β - 2) a_{0} - 2 a_{1}}{2 (α + β)} \int_{Ω} (1 + u_{n}^{2}) {| \nabla u_{n} |}^{2} \\ + \frac{(α + β - 2) b_{0} - 2 b_{1}}{2 (α + β)} \int_{Ω} (1 + v_{n}^{2}) {| \nabla v_{n} |}^{2} \\ \geq (\frac{1}{4} - \frac{1}{α + β}) μ \int_{Ω} ({| \nabla u |}^{4} + {| \nabla v |}^{4}) . \end{aligned}

Thus we have

\underset{n \to \infty}{lim sup} {∥ (u_{n}, v_{n}) ∥}^{4} \leq {(\frac{1}{4} - \frac{1}{α + β})}^{- 1} μ^{- 1} c .

This completes the proof of Lemma 3.2. □

Now we give the proof of Proposition 3.1.

Proof of Proposition 3.1 From Lemma 3.2 , we know that

{(u_{n}, v_{n})}

is bounded in

W_{0}^{1, 4} (Ω) \times W_{0}^{1, 4} (Ω)

. So there exists a subsequence of

{(u_{n}, v_{n})}

, still denoted by

{(u_{n}, v_{n})}

, such that

\begin{matrix} (u_{n}, v_{n}) ⇀ (u, v) weakly in W_{0}^{1, 4} (Ω) \times W_{0}^{1, 4} (Ω) as n \to \infty, \\ u_{n} \to u, v_{n} \to v strongly in L^{s} (Ω) as n \to \infty for any 2 < s < 2 \cdot 2^{*} . \end{matrix}

Now we prove that

(u_{n}, v_{n}) \to (u, v)

in

W_{0}^{1, 4} (Ω) \times W_{0}^{1, 4} (Ω)

. In (2.5), choosing

(φ, ψ) = (u_{n} - u_{m}, v_{n} - v_{m})

, we have

\begin{aligned} o (1) ∥ (u_{n} - u_{m}, v_{n} - v_{m}) ∥ \\ = 〈 I_{μ}^{'} (u_{n}, v_{n}) - I_{μ}^{'} (u_{m}, v_{m}), (u_{n} - u_{m}, v_{n} - v_{m}) 〉 \\ = μ \int_{Ω} ({| \nabla u_{n} |}^{2} \nabla u_{n} - {| \nabla u_{m} |}^{2} \nabla u_{m}) (\nabla u_{n} - \nabla u_{m}) \\ + μ \int_{Ω} ({| \nabla v_{n} |}^{2} \nabla v_{n} - {| \nabla v_{m} |}^{2} \nabla v_{m}) (\nabla v_{n} - \nabla v_{m}) \\ + \int_{Ω} \sum_{i, j = 1}^{N} (a_{i j} (x, u_{n}) D_{i} u_{n} - a_{i j} (x, u_{m}) D_{i} u_{m}) (D_{j} u_{n} - D_{j} u_{m}) \\ + \int_{Ω} \sum_{i, j = 1}^{N} (b_{i j} (x, v_{n}) D_{i} v_{n} - b_{i j} (x, v_{m}) D_{i} v_{m}) (D_{j} v_{n} - D_{j} v_{m}) \\ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} (D_{s} a_{i j} (x, u_{n}) D_{i} u_{n} D_{j} u_{n} - D_{s} a_{i j} (x, u_{m}) D_{i} u_{m} D_{j} u_{m}) (u_{n} - u_{m}) \\ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} (D_{s} b_{i j} (x, v_{n}) D_{i} v_{n} D_{j} v_{n} - D_{s} b_{i j} (x, v_{m}) D_{i} v_{m} D_{j} v_{m}) (v_{n} - v_{m}) \\ - \frac{2 α}{α + β} \int_{Ω} ({| u_{n} |}^{α - 2} {| v_{n} |}^{β} u_{n} - {| u_{m} |}^{α - 2} {| v_{m} |}^{β} u_{m}) (u_{n} - u_{m}) \\ - \frac{2 β}{α + β} \int_{Ω} ({| u_{n} |}^{α} {| v_{n} |}^{β - 2} v_{n} - {| u_{m} |}^{α} {| v_{m} |}^{β - 2} v_{m}) (v_{n} - v_{m}) . \end{aligned}

(3.3)

We may estimate the terms involved as follows:

\begin{aligned} μ \int_{Ω} ({| \nabla u_{n} |}^{2} \nabla u_{n} - {| \nabla u_{m} |}^{2} \nabla u_{m}) (\nabla u_{n} - \nabla u_{m}) \geq \frac{1}{4} μ \int_{Ω} {| \nabla u_{n} - \nabla u_{m} |}^{4}, \\ μ \int_{Ω} ({| \nabla v_{n} |}^{2} \nabla v_{n} - {| \nabla v_{m} |}^{2} \nabla v_{m}) (\nabla v_{n} - \nabla v_{m}) \geq \frac{1}{4} μ \int_{Ω} {| \nabla v_{n} - \nabla v_{m} |}^{4}, \\ \int_{Ω} \sum_{i, j = 1}^{N} (a_{i j} (x, u_{n}) D_{i} u_{n} - a_{i j} (x, u_{m}) D_{i} u_{m}) (D_{j} u_{n} - D_{j} u_{m}) \\ = \int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, u_{n}) D_{i} (u_{n} - u_{m}) D_{j} (u_{n} - u_{m}) \\ + \int_{Ω} \sum_{i, j = 1}^{N} (a_{i j} (x, u_{n}) - a_{i j} (x, u_{m})) D_{i} u_{m} D_{j} (u_{n} - u_{m}) \\ \geq \int_{Ω} \sum_{i, j = 1}^{N} D_{s} a_{i j} (x, t u_{n} + (1 - t) u_{m}) (u_{n} - u_{m}) D_{i} u_{m} D_{j} (u_{n} - u_{m}) \\ \geq - C {| u_{n} - u_{m} |}_{4} ∥ u_{m} ∥ (∥ u_{n} ∥ + ∥ u_{m} ∥) \\ \to 0 as m, n \to \infty for some t \in (0, 1), \\ \int_{Ω} \sum_{i, j = 1}^{N} (b_{i j} (x, v_{n}) D_{i} v_{n} - b_{i j} (x, v_{m}) D_{i} v_{m}) (D_{j} v_{n} - D_{j} v_{m}) \\ = \int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, v_{n}) D_{i} (v_{n} - v_{m}) D_{j} (v_{n} - v_{m}) \\ + \int_{Ω} \sum_{i, j = 1}^{N} (b_{i j} (x, v_{n}) - b_{i j} (x, v_{m})) D_{i} v_{m} D_{j} (v_{n} - v_{m}) \\ \geq \int_{Ω} \sum_{i, j = 1}^{N} D_{s} b_{i j} (x, τ v_{n} + (1 - τ) v_{m}) (v_{n} - v_{m}) D_{i} v_{m} D_{j} (v_{n} - v_{m}) \\ \geq - C {| v_{n} - v_{m} |}_{4} ∥ v_{m} ∥ (∥ v_{n} ∥ + ∥ v_{m} ∥) \\ \to 0 as m, n \to \infty for some τ \in (0, 1), \\ \frac{1}{2} | \int_{Ω} \sum_{i, j = 1}^{N} (D_{s} a_{i j} (x, u_{n}) D_{i} u_{n} D_{j} u_{n} - D_{s} a_{i j} (x, u_{m}) D_{i} u_{m} D_{j} u_{m}) (u_{n} - u_{m}) | \\ \leq \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} | D_{s} a_{i j} (x, u_{n}) D_{i} u_{n} D_{j} u_{n} (u_{n} - u_{m}) | \\ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} | D_{s} a_{i j} (x, u_{n}) D_{i} u_{n} D_{j} u_{n} (u_{n} - u_{m}) | \\ \leq C ({∥ u_{n} ∥}^{2} + {∥ u_{m} ∥}^{2}) {| u_{n} - u_{m} |}_{4} \\ \to 0 as m, n \to \infty, \\ \frac{1}{2} | \int_{Ω} \sum_{i, j = 1}^{N} (D_{s} b_{i j} (x, v_{n}) D_{i} v_{n} D_{j} v_{n} - D_{s} b_{i j} (x, v_{m}) D_{i} v_{m} D_{j} v_{m}) (v_{n} - v_{m}) | \\ \leq \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} | D_{s} b_{i j} (x, v_{n}) D_{i} v_{n} D_{j} v_{n} (v_{n} - v_{m}) | \\ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} | D_{s} b_{i j} (x, v_{n}) D_{i} v_{n} D_{j} v_{n} (v_{n} - v_{m}) | \\ \leq C ({∥ v_{n} ∥}^{2} + {∥ v_{m} ∥}^{2}) {| v_{n} - v_{m} |}_{4} \\ \to 0 as m, n \to \infty, \\ \frac{2 α}{α + β} | \int_{Ω} ({| u_{n} |}^{α - 2} {| v_{n} |}^{β} u_{n} - {| u_{m} |}^{α - 2} {| v_{m} |}^{β} u_{m}) (u_{n} - u_{m}) | \\ \leq \frac{2 α}{α + β} \int_{Ω} ({| u_{n} |}^{α - 1} {| v_{n} |}^{β} + {| u_{m} |}^{α - 1} {| v_{m} |}^{β}) | u_{n} - u_{m} | \\ \leq \frac{2 α}{α + β} ({| u_{n} |}_{α + β}^{α - 1} {| v_{n} |}_{α + β}^{β} + {| u_{m} |}_{α + β}^{α - 1} {| v_{m} |}_{α + β}^{β}) {| u_{n} - u_{m} |}_{α + β} \\ \to 0 as m, n \to \infty, \\ \frac{2 β}{α + β} | \int_{Ω} ({| u_{n} |}^{α} {| v_{n} |}^{β - 2} v_{n} - {| u_{m} |}^{α} {| v_{m} |}^{β - 2} v_{m}) (v_{n} - v_{m}) | \\ \leq \frac{2 β}{α + β} \int_{Ω} ({| u_{n} |}^{α} {| v_{n} |}^{β - 1} + {| u_{m} |}^{α} {| v_{m} |}^{β - 1}) | v_{n} - v_{m} | \\ \leq \frac{2 β}{α + β} ({| u_{n} |}_{α + β}^{α} {| v_{n} |}_{α + β}^{β - 1} + {| u_{m} |}_{α + β}^{α} {| v_{m} |}_{α + β}^{β - 1}) {| v_{n} - v_{m} |}_{α + β} \\ \to 0 as m, n \to \infty . \end{aligned}

Returning to (3.3), we have

\frac{1}{4} μ \int_{Ω} ({| \nabla u_{n} - \nabla u_{m} |}^{4} + {| \nabla v_{n} - \nabla v_{m} |}^{4}) \leq o (1) ∥ (u_{n} - u_{m}, v_{n} - v_{m}) ∥ + o (1),

which implies that $∥ (u_{n} - u_{m}, v_{n} - v_{m}) ∥ \to 0$ , i.e., $(u_{n}, u_{m}) \to (u, v)$ in $W_{0}^{1, 4} (Ω) \times W_{0}^{1, 4} (Ω)$ . This completes the proof of Proposition 3.1. □

4 Some asymptotic behavior

Proposition 3.1 enables us to apply minimax argument to the functional

I_{μ} (u, v)

. In this section, we also study the behavior of the sequences

{(u_{n}, v_{n})} \subset W_{0}^{1, 4} (Ω) \times W_{0}^{1, 4} (Ω)

and

{μ_{n}} \subset (0, 1]

satisfying

μ_{n} \to 0,

(4.1)

I_{μ_{n}} (u_{n}, v_{n}) \to c,

(4.2)

{∥ I_{μ_{n}}^{'} (u_{n}, v_{n}) ∥}^{*} \to 0 .

(4.3)

The following proposition is the key of this section.

Proposition 4.1 Assume that the sequences

{(u_{n}, v_{n})} \subset W_{0}^{1, 4} (Ω) \times W_{0}^{1, 4} (Ω)

and

{μ_{n}} \subset (0, 1]

satisfy (4.1)-(4.3). Then, after extracting a sequence, still denoted by n, we have

\begin{matrix} (u_{n}, v_{n}) ⇀ (u, v) in W_{0}^{1, 2} (Ω) \times W_{0}^{1, 2} (Ω), \\ (u_{n} \nabla u_{n}, v_{n} \nabla v_{n}) ⇀ (u \nabla u, v \nabla v) in L^{2} (Ω) \times L^{2} (Ω) \end{matrix}

and

(u_{n} (x), v_{n} (x)) \to (u (x), v (x)) a.e. x \in Ω

as $n \to \infty$ .

Proof Similar to the proof of Lemma 3.2, by (4.1)-(4.3), we have

\begin{array}{rcl} C & \geq & I_{μ_{n}} (u_{n}, v_{n}) - \frac{1}{α + β} 〈 I_{μ_{n}}^{'} (u_{n}, v_{n}), (u_{n}, v_{n}) 〉 \\ \geq & (\frac{1}{4} - \frac{1}{α + β}) μ_{n} \int_{Ω} ({| \nabla u_{n} |}^{4} + {| \nabla v_{n} |}^{4}) \\ + \frac{(α + β - 2) a_{0} - 2 a_{1}}{2 (α + β)} \int_{Ω} (1 + u_{n}^{2}) {| \nabla u_{n} |}^{2} \\ + \frac{(α + β - 2) b_{0} - 2 b_{1}}{2 (α + β)} \int_{Ω} (1 + v_{n}^{2}) {| \nabla v_{n} |}^{2} . \end{array}

(4.4)

Thus

μ_{n} \int_{Ω} ({| \nabla u_{n} |}^{4} + {| \nabla v_{n} |}^{4}) + \int_{Ω} ({| \nabla u_{n} |}^{2} + {| \nabla v_{n} |}^{2}) + \int_{Ω} (u_{n}^{2} {| \nabla u_{n} |}^{2} + v_{n}^{2} {| \nabla v_{n} |}^{2}) \leq C

(4.5)

for some C independent of n. Then, up to a subsequence, we have

\begin{matrix} (u_{n}, v_{n}) ⇀ (u, v) in W_{0}^{1, 2} (Ω) \times W_{0}^{1, 2} (Ω), \\ (u_{n} \nabla u_{n}, v_{n} \nabla v_{n}) ⇀ (u \nabla u, v \nabla v) in L^{2} (Ω) \times L^{2} (Ω) \end{matrix}

and

(u_{n} (x), v_{n} (x)) \to (u (x), v (x)) a.e. x \in Ω

as $n \to \infty$ . This completes the proof of Proposition 4.1. □

5 Proof of main results

In this section, we give the proof of our main results. Firstly, we prove Theorem 2.1.

Proof of Theorem 2.1 Note that

(u_{n}, v_{n})

satisfies the following equation:

\begin{aligned} μ_{n} \int_{Ω} {| \nabla u_{n} |}^{2} \nabla u_{n} \nabla φ + μ_{n} \int_{Ω} {| \nabla v_{n} |}^{2} \nabla v_{n} \nabla ψ \\ + \int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, u_{n}) D_{i} u_{n} D_{j} φ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} D_{s} a_{i j} (x, u_{n}) D_{i} u_{n} D_{j} u_{n} φ \\ + \int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, v_{n}) D_{i} v_{n} D_{j} ψ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} D_{s} b_{i j} (x, v_{n}) D_{i} v_{n} D_{j} v_{n} ψ \\ - \frac{2 α}{α + β} \int_{Ω} {| u_{n} |}^{α - 2} {| v_{n} |}^{β} u_{n} φ - \frac{2 β}{α + β} \int_{Ω} {| u_{n} |}^{α} {| v_{n} |}^{β - 2} v_{n} ψ = 0 \end{aligned}

(5.1)

for all

(φ, ψ) \in W_{0}^{1, 4} (Ω) \times W_{0}^{1, 4} (Ω)

. Since

{(\int_{Ω} {| u_{n} |}^{\frac{4 N}{N - 2}})}^{\frac{N - 2}{N}} \leq C \int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, u_{n}) D_{i} u_{n} D_{j} u_{n} \leq C

and

{(\int_{Ω} {| v_{n} |}^{\frac{4 N}{N - 2}})}^{\frac{N - 2}{N}} \leq C \int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, v_{n}) D_{i} v_{n} D_{j} v_{n} \leq C .

By Moser’s iteration, we have

{| u_{n} |}_{L^{\infty}} \leq C, {| v_{n} |}_{L^{\infty}} \leq C .

(5.2)

Hence

{| u |}_{L^{\infty}} \leq C, {| v |}_{L^{\infty}} \leq C

(5.3)

for some C independent of n. To show that

(u, v)

is a critical point of

I_{0}

, we use some arguments in [22, 23] (see more references therein). In (5.1), we choose

φ = ξ exp (- M u_{n})

,

ψ = η exp (- M v_{n})

, where

ξ \in C_{0}^{\infty} (Ω)

,

ξ \geq 0

,

η \in C_{0}^{\infty} (Ω)

,

η \geq 0

and

M > 0

is a constant. Substituting

(φ, ψ)

into (5.1), we have

\begin{array}{rcl} 0 & = & μ_{n} \int_{Ω} {| \nabla u_{n} |}^{2} \nabla u_{n} (\nabla ξ exp (- M u_{n}) - ξ \nabla u_{n} exp (- M u_{n})) \\ + μ_{n} \int_{Ω} {| \nabla v_{n} |}^{2} \nabla v_{n} (\nabla η exp (- M v_{n}) - η \nabla v_{n} exp (- M v_{n})) \\ + \int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, u_{n}) D_{i} u_{n} (D_{j} ξ exp (- M u_{n}) - M ξ D_{j} u_{n} exp (- M u_{n})) \\ + \int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, v_{n}) D_{i} v_{n} (D_{j} η exp (- M v_{n}) - M η D_{j} v_{n} exp (- M v_{n})) \\ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} D_{s} a_{i j} (x, u_{n}) D_{i} u_{n} D_{j} u_{n} ξ exp (- M u_{n}) \\ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} D_{s} b_{i j} (x, v_{n}) D_{i} v_{n} D_{j} v_{n} η exp (- M v_{n}) \\ - \frac{2 α}{α + β} \int_{Ω} {| u_{n} |}^{α - 2} {| v_{n} |}^{β} u_{n} ξ exp (- M u_{n}) - \frac{2 β}{α + β} \int_{Ω} {| u_{n} |}^{α} {| v_{n} |}^{β - 2} v_{n} η exp (- M v_{n}) \\ \leq & μ_{n} \int_{Ω} {| \nabla u_{n} |}^{2} \nabla u_{n} \nabla ξ exp (- M u_{n}) + μ_{n} \int_{Ω} {| \nabla v_{n} |}^{2} \nabla v_{n} \nabla η exp (- M v_{n}) \\ + \int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, u_{n}) D_{i} u_{n} D_{j} ξ exp (- M u_{n}) + \int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, v_{n}) D_{i} v_{n} D_{j} η exp (- M v_{n}) \\ - \int_{Ω} \sum_{i, j = 1}^{N} (M a_{i j} (x, u_{n}) - \frac{1}{2} D_{s} a_{i j} (x, u_{n})) D_{i} u_{n} D_{j} u_{n} ξ exp (- M u_{n}) \\ - \int_{Ω} \sum_{i, j = 1}^{N} (M b_{i j} (x, v_{n}) - \frac{1}{2} D_{s} b_{i j} (x, v_{n})) D_{i} v_{n} D_{j} v_{n} η exp (- M v_{n}) \\ - \frac{2 α}{α + β} \int_{Ω} {| u_{n} |}^{α - 2} {| v_{n} |}^{β} u_{n} ξ exp (- M u_{n}) \\ - \frac{2 β}{α + β} \int_{Ω} {| u_{n} |}^{α} {| v_{n} |}^{β - 2} v_{n} η exp (- M v_{n}) . \end{array}

(5.4)

Note that

M a_{i j} (x, u_{n}) - \frac{1}{2} D_{s} a_{i j} (x, u_{n})

,

M b_{i j} (x, v_{n}) - \frac{1}{2} D_{s} b_{i j} (x, v_{n})

are positive for M large enough. By Fatou’s lemma, the weak convergence of

{(u_{n}, v_{n})}

and the fact that

μ_{n} \int_{Ω} ({| \nabla u_{n} |}^{4} + {| \nabla v_{n} |}^{4})

is bounded, we have

\begin{array}{rcl} 0 & \leq & \int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, u) D_{i} u D_{j} ξ exp (- M u) + \int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, v) D_{i} v D_{j} η exp (- M v) \\ - \int_{Ω} \sum_{i, j = 1}^{N} (M a_{i j} (x, u) - \frac{1}{2} D_{s} a_{i j} (x, u)) D_{i} u D_{j} u ξ exp (- M u) \\ - \int_{Ω} \sum_{i, j = 1}^{N} (M b_{i j} (x, v) - \frac{1}{2} D_{s} b_{i j} (x, v)) D_{i} v D_{j} v η exp (- M v) \\ - \frac{2 α}{α + β} \int_{Ω} {| u |}^{α - 2} {| v |}^{β} u ξ exp (- M u) - \frac{2 β}{α + β} \int_{Ω} {| u |}^{α} {| v |}^{β - 2} v η exp (- M v) \\ = & \int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, u) D_{i} u D_{j} (ξ exp (- M u)) + \int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, u) D_{i} v D_{j} (η exp (- M v)) \\ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} D_{s} a_{i j} (x, u) D_{i} u D_{j} u ξ exp (- M u) \\ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} D_{s} b_{i j} (x, v) D_{i} v D_{j} v η exp (- M v) \\ - \frac{2 α}{α + β} \int_{Ω} {| u |}^{α - 2} {| v |}^{β} u ξ exp (- M u) - \frac{2 β}{α + β} \int_{Ω} {| u |}^{α} {| v |}^{β - 2} v η exp (- M v) . \end{array}

(5.5)

Let

(χ, ω) \geq (0, 0)

,

(χ, ω) \in C_{0}^{\infty} (Ω) \times C_{0}^{\infty} (Ω)

. We may choose

ξ = χ exp (M u)

,

η = ω exp (M v)

such that

(ξ, η) \in W_{0}^{1, 4} (Ω) \times W_{0}^{1, 4} (Ω)

,

{| ξ |}_{L^{\infty} (Ω)} \leq C

and

{| η |}_{L^{\infty} (Ω)} \leq C

. Then we obtain

\begin{aligned} \int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, u) D_{i} u D_{j} χ + \int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, v) D_{i} v D_{j} ω \\ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} D_{s} a_{i j} (x, u) D_{i} u D_{j} u χ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} D_{s} b_{i j} (x, v) D_{i} v D_{j} v ω \\ - \frac{2 α}{α + β} \int_{Ω} {| u |}^{α - 2} {| v |}^{β} u χ - \frac{2 β}{α + β} \int_{Ω} {| u |}^{α} {| v |}^{β - 2} v ω \geq 0 \end{aligned}

(5.6)

for all $(χ, ω) \geq (0, 0)$ , $(χ, ω) \in C_{0}^{\infty} (Ω) \times C_{0}^{\infty} (Ω)$ .

Similarly, we may obtain an opposite inequality. Thus we have

\begin{aligned} \int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, u) D_{i} u D_{j} χ + \int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, v) D_{i} v D_{j} ω \\ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} D_{s} a_{i j} (x, u) D_{i} u D_{j} u χ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} D_{s} b_{i j} (x, v) D_{i} v D_{j} v ω \\ - \frac{2 α}{α + β} \int_{Ω} {| u |}^{α - 2} {| v |}^{β} u χ - \frac{2 β}{α + β} \int_{Ω} {| u |}^{α} {| v |}^{β - 2} v ω = 0 \end{aligned}

(5.7)

for all

(χ, ω) \in C_{0}^{\infty} (Ω) \times C_{0}^{\infty} (Ω)

. That is,

(u, v)

is a critical point of

I_{0}

and a solution for system (1.1). By doing approximations, we have

(u, v)

in the place of

(χ, ω)

of (5.7)

\begin{aligned} \int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, u) D_{i} u D_{j} u + \int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, v) D_{i} v D_{j} v \\ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} D_{s} a_{i j} (x, u) u D_{i} u D_{j} u \\ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} D_{s} b_{i j} (x, v) v D_{i} v D_{j} v - 2 \int_{Ω} {| u |}^{α} {| v |}^{β} = 0 . \end{aligned}

(5.8)

Setting

(φ, ψ) = (u_{n}, v_{n})

in (5.1), we have

\begin{aligned} μ_{n} \int_{Ω} ({| \nabla u_{n} |}^{4} + {| \nabla v_{n} |}^{4}) + \int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, u_{n}) D_{i} u_{n} D_{j} u_{n} \\ + \int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, v_{n}) D_{i} v_{n} D_{j} v_{n} + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} D_{s} a_{i j} (x, u_{n}) u_{n} D_{i} u_{n} D_{j} u_{n} \\ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} D_{s} b_{i j} (x, v_{n}) v_{n} D_{i} v_{n} D_{j} v_{n} - 2 \int_{Ω} {| u_{n} |}^{α} {| v_{n} |}^{β} = 0 . \end{aligned}

(5.9)

Using

\int_{Ω} {| u_{n} |}^{α} {| v_{n} |}^{β} \to \int_{Ω} {| u |}^{α} {| v |}^{β}

as

n \to \infty

, (5.8), (5.9) and lower semi-continuity, we obtain

\begin{matrix} μ_{n} \int_{Ω} ({| \nabla u_{n} |}^{4} + {| \nabla v_{n} |}^{4}) \to 0, \\ \int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, u_{n}) D_{i} u_{n} D_{j} u_{n} \to \int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, u) D_{i} u D_{j} u, \\ \int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, v_{n}) D_{i} v_{n} D_{j} v_{n} \to \int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, v) D_{i} v D_{j} v \end{matrix}

as $n \to \infty$ .

In particular, we have

\begin{matrix} u_{n} \to u, v_{n} \to v in W_{0}^{1, 2} (Ω), \\ u_{n} \nabla u_{n} \to u \nabla u, v_{n} \nabla v_{n} \to v \nabla v in L^{2} (Ω) \end{matrix}

and

I_{μ_{n}}^{'} (u_{n}, v_{n}) \to I_{0}^{'} (u, v)

as $n \to \infty$ . This completes the proof of Theorem 2.1. □

Next, we apply the mountain pass theorem to obtain the existence of critical points of

I_{μ}

. Set

\begin{aligned} Σ_{ρ} = & {(u, v) \in W_{0}^{1, 4} (Ω) \times W_{0}^{1, 4} (Ω) | \\ \int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, u) D_{i} u D_{j} u + \int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, v) D_{i} v D_{j} v \leq ρ^{2}} \end{aligned}

for $ρ > 0$ .

Let us consider the functional

\begin{array}{rcl} I_{μ}^{+} (u, v) & = & \frac{1}{4} μ \int_{Ω} ({| \nabla u |}^{4} + {| \nabla v |}^{4}) + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, u) D_{i} u D_{j} u \\ + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, v) D_{i} v D_{j} v - \frac{2}{α + β} \int_{Ω} {(u^{+})}^{α} {(v^{+})}^{β} . \end{array}

(5.10)

Here and in what follows, we denote

u^{+} = max {u, 0}

. The functional

I_{μ}

satisfies

{(PS)}_{c}

condition. Similarly, we may verify that

I_{μ}^{+}

satisfies

{(PS)}_{c}

condition. By the ε-Young inequality, for any

ε > 0

, there exists

C_{ε} > 0

such that

{(u^{+})}^{α} {(v^{+})}^{β} \leq ε {(u^{+})}^{α + β} + C_{ε} {(v^{+})}^{α + β}

and

\begin{matrix} \int_{Ω} {| u |}^{α + β} \leq C {(\int_{Ω} u^{2} {| \nabla u |}^{2})}^{\frac{α + β}{4}} \leq C {(\int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, u) D_{i} u D_{j} u)}^{\frac{α + β}{4}}, \\ \int_{Ω} {| v |}^{α + β} \leq C {(\int_{Ω} v^{2} {| \nabla v |}^{2})}^{\frac{α + β}{4}} \leq C {(\int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, v) D_{i} v D_{j} v)}^{\frac{α + β}{4}} . \end{matrix}

Then

\begin{aligned} - \frac{2}{α + β} \int_{Ω} {(u^{+})}^{α} {(v^{+})}^{β} \\ \geq - \frac{2}{α + β} ε \int_{Ω} {(u^{+})}^{α + β} - \frac{2}{α + β} C_{ε} \int_{Ω} {(u^{+})}^{α + β} \\ \geq - \frac{2 C}{α + β} ε {(\int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, u) D_{i} u D_{j} u)}^{\frac{α + β}{4}} - \frac{2 C_{ε}}{α + β} {(\int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, v) D_{i} v D_{j} v)}^{\frac{α + β}{4}} \\ \geq - \frac{2 C}{α + β} ε ρ^{\frac{α + β}{2}} - \frac{2 C_{ε}}{α + β} ρ^{\frac{α + β}{2}} \\ \geq - \frac{1}{α + β} ρ^{2} \end{aligned}

for ε, ρ small. Thus we have

\begin{array}{rcl} I_{μ}^{+} (u, v) & \geq & \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, u) D_{i} u D_{j} u + \frac{1}{2} \int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, v) D_{i} v D_{j} v - \frac{2}{α + β} \int_{Ω} {(u^{+})}^{α} {(v^{+})}^{β} \\ \geq & \frac{1}{2} ρ^{2} - \frac{1}{α + β} ρ^{2} = (\frac{1}{2} - \frac{1}{α + β}) ρ^{2} \end{array}

for

(u, v) \in \partial Σ_{ρ}

and for

ρ > 0

small enough. Choose

(φ, ψ) \geq (0, 0)

,

(χ, ω) \in C_{0}^{\infty} (Ω) \times C_{0}^{\infty} (Ω)

and

T > 0

. Define a path

(g, h) : [0, 1] \to W_{0}^{1, 4} (Ω) \times W_{0}^{1, 4} (Ω)

by

(g (t), h (t)) = (t T φ, t T ψ)

. When T is large enough, we have

\begin{matrix} I_{μ}^{+} (g (1), h (1)) < 0, \\ \int_{Ω} \sum_{i, j = 1}^{N} a_{i j} (x, g (1)) D_{i} g (1) D_{j} g (1) + \int_{Ω} \sum_{i, j = 1}^{N} b_{i j} (x, h (1)) D_{i} h (1) D_{j} h (1) > ρ^{2} \end{matrix}

and

sup_{t \in [0, 1]} I_{μ}^{+} (g (t), h (t)) \leq m

for some m independent of $μ \in (0, 1]$ .

Define

c_{μ} = inf_{(g, h) \in Γ} sup_{t \in [0, 1]} I_{μ}^{+} (g (t), h (t)),

where

\begin{aligned} Γ = & {(g, h) \in C ([0, 1], W_{0}^{1, 4} (Ω) \times W_{0}^{1, 4} (Ω)) | \\ (g (0), h (0)) = (0, 0), (g (1), h (1)) = (T φ, T ψ)} . \end{aligned}

From the mountain pass theorem we obtain that

c_{μ} \geq (\frac{1}{2} - \frac{1}{α + β}) ρ^{2}

is a critical value of $I_{μ}^{+}$ .

Let $(u_{μ}, v_{μ})$ be a critical point corresponding to $c_{μ}$ . We have $(u_{μ}, v_{μ}) \geq (0, 0)$ . Thus $(u_{μ}, v_{μ})$ is a positive critical point of $I_{μ}$ by the strong maximum principle. In summary, we have the following.

Proposition 5.1 There exist positive constants ρ and m independent of μ such that

I_{μ}

has a positive critical point

(u_{μ}, v_{μ})

satisfying

(\frac{1}{2} - \frac{1}{α + β}) ρ^{2} \leq I_{μ} (u_{μ}, v_{μ}) \leq m .

Finally, we give the proof of Theorem 2.2.

Proof of Theorem 2.2 For a positive solution of system (1.1), the proof follows from Proposition 5.1 and Theorem 2.1. A similar argument gives a negative solution of system (1.1). This completes the proof of Theorem 2.2. □

Declarations

Acknowledgements

This paper was finished while the first author was a visiting fellow at the School of Mathematical Sciences of Beijing Normal University, and the first author would like to express her gratitude for their hospitality during her visit. This work is supported by the National Science Foundation of China (11061031), Fundamental Research Funds for the Central Universities (31920130004) and Fundamental Research Funds for the Gansu University.

Authors’ Affiliations

(1)

College of Mathematics and Computer Science, Northwest University for Nationalities

(2)

College of Mathematics and Statistics, Northwest Normal University

(3)

School of Mathematical Sciences, Beijing Normal University

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