# Existence of multiple solutions for fractional p-Kirchhoff equations with concave-convex nonlinearities

## Abstract

In this paper, we investigate the existence of multiple solutions for Kirchhoff-type equations involving nonlocal integro-differential operators with homogeneous Dirichlet boundary conditions as follows:

$$\textstyle\begin{cases} M(\int_{\mathbb{R}^{2n}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,dx\, dy)(-\triangle )_{p}^{s}u=\lambda|u|^{q-2}u+\frac{\alpha}{\alpha+\beta}|u|^{\alpha -2}u|v|^{\beta}, & \mbox{in }\Omega, \\ M(\int_{\mathbb{R}^{2n}}\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}\,dx\, dy)(-\triangle )_{p}^{s}v=\mu|v|^{q-2}v+\frac{\beta}{\alpha+\beta}|v|^{\beta -2}v|u|^{\alpha}, & \mbox{in }\Omega, \\ u=v=0, & \mbox{in }\mathbb{R}^{n}\setminus\Omega, \end{cases}$$

where Ω is a smooth bounded set in $$\mathbb{R}^{n}$$, $$n>ps$$ with $$s\in(0,1)$$ fixed, $${\lambda,\mu}>0$$ are two parameters, $$1< q< p< p(\tau+1)<\alpha+\beta<p^{*}$$, $$p^{*}=\frac{np}{n-sp}$$, M is a continuous function, given by $$M(h)=k+lh^{\tau}$$, $$k>0$$, $$l,\tau\geq0$$, and $$(-\triangle)_{p}^{s}$$ is the fractional p-Laplacian operator. We will prove that the problem has at least two solutions by using the Nehari manifold method and fibering maps.

## Introduction

In this paper, we consider the following Kirchhoff-type problem involving fractional p-Laplacian and concave-convex nonlinearities:

$$\textstyle\begin{cases} M(\int_{\mathbb{R}^{2n}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,dx\, dy)(-\triangle )_{p}^{s}u=\lambda|u|^{q-2}u+\frac{\alpha}{\alpha+\beta}|u|^{\alpha -2}u|v|^{\beta}, & \mbox{in }\Omega, \\ M(\int_{\mathbb{R}^{2n}}\frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}\,dx\, dy)(-\triangle )_{p}^{s}v=\mu|v|^{q-2}v+\frac{\beta}{\alpha+\beta}|v|^{\beta -2}v|u|^{\alpha}, & \mbox{in }\Omega, \\ u=v=0,& \mbox{in }\mathbb{R}^{n}\setminus\Omega, \end{cases}$$
(1.1)

where Ω is a smooth bounded set in $$\mathbb{R}^{n}$$, $$n>ps$$ with $$s\in (0,1)$$ fixed, $${\lambda,\mu}>0$$ are two parameters, $$1< q< p< p(\tau+1)<\alpha+\beta<p^{*}$$, $$p^{*}=\frac{np}{n-sp}$$ is the fractional Sobolev exponent, M is a special continuous function defined by $$M(h)=k+lh^{\tau}$$, $$k>0$$, $$l,\tau\geq0$$. $$(-\triangle)_{p}^{s}$$ is the fractional p-Laplacian operator given by

$$(-\triangle)_{p}^{s}u(x)=2\lim _{\varepsilon\rightarrow0} \int_{\mathbb{R} ^{n}\setminus {B_{\varepsilon}(x)}} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{n+sp}}\,dx\,dy.$$
(1.2)

The Kirchhoff-type equation and system have a broad background in phase transitions, population dynamics, mathematical finance, etc. There have been a lot of excellent results related to the existence and multiplicity of solutions for this system. We refer the readers to  for Kirchhoff problems involving the classical Laplace operator and to [5, 6] for the p-Laplacian case. For the fractional system, please consult  and the references therein.

In  and , the authors discussed the system (or a single equation, that is, $$u=v$$) in the special case of $$M\equiv 1$$. They obtained some interesting results by using the Nehari manifold method. For the special case $$p=2$$ of this system, there are many results available in the existing literature, we refer the interested reader to [22, 23] for the case of the classical Laplacian and to  for the case of the fractional Laplacian. Moreover, the authors  studied bifurcation results for a fractional elliptic equation with critical exponent. There is also some work for the case that M is not a constant (see, for example, ). However, as far as we know, there are few results on the fractional p-Kirchhoff system with concave-convex nonlinearities. Motivated by the above work, in this paper we consider problem (1.1) for a more general case $$M(h)=k+lh^{\tau}$$. We obtained a new multiplicity result by using the Nehari manifold method and fibering maps.

In order to state our result, we introduce some notations. Suppose $$s\in (0,1)$$ and $$p\in[1,\infty)$$. Let $$W^{s,p}$$ be a fractional Sobolev space with the norm

$$\|u\|_{W^{s,p} (\Omega)}= \|u\|_{L^{p}(\Omega)}+ \biggl( \int_{\Omega \times \Omega} \frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,dx\,dy \biggr)^{\frac{1}{p}}.$$
(1.3)

Set $$Q=\mathbb{R}^{2n}\setminus(C\Omega\times{C\Omega})$$ with $$C\Omega=\mathbb{R}^{n}\setminus\Omega$$. We define

$$X = \biggl\{ u\Big|u:\mathbb{R}^{n}\rightarrow\mathbb{R}\mbox{ is measurable}, u|_{\Omega}\in L^{p}(\Omega),\mbox{and } \int_{Q}\frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,dx\,dy< \infty\biggr\} .$$

The space X is endowed with the norm

$$\|u\|_{X}= \|u\|_{L^{p}(\Omega)}+ \biggl( \int_{Q} \frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,dx\,dy \biggr)^{\frac{1}{p}}.$$
(1.4)

Let $$X_{0}$$ be the completion of the space $$C_{0}^{\infty}(\Omega)$$ in X. The space $$X_{0}$$ is a Banach space which can be endowed with the norm

$$\|u\|_{X_{0}}= \biggl( \int_{Q} \frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,dx\,dy \biggr)^{\frac{1}{p}}.$$
(1.5)

It is easy to see that this norm is equivalent to the usual one defined in (1.3).

As proved in [17, 24], we have the following results:

1. (i)

$$X_{0}\hookrightarrow L^{r}(\Omega)$$ is continuous for any $$r\in [1,p^{*}]$$ and compact for any $$r\in[1,p^{*})$$.

2. (ii)

For $$\alpha+\beta\in(p,p^{*})$$, let S denote the best Sobolev constant for the embedding $$X_{0}\hookrightarrow L^{\alpha+\beta}(\Omega)$$. Then, for $$u\in X_{0}$$, we have

\begin{aligned} \begin{aligned}[b] \|u\|_{L^{\alpha+\beta}(\Omega)}&= \biggl( \int_{\Omega}|u|^{\alpha +\beta }\,dx \biggr)^{\frac{1}{ \alpha+\beta}} \leq S^{-\frac{1}{p}}\|u\|_{X_{0}} \\ &=S^{-\frac{1}{p}} \biggl( \int_{Q} \frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,dx\,dy \biggr)^{\frac{1}{p}}. \end{aligned} \end{aligned}
(1.6)

Let $$E=X_{0}\times X_{0}$$ be the Cartesian product of two spaces, which is a reflexive Banach space with the norm

\begin{aligned} \bigl\Vert (u,v)\bigr\Vert =&\bigl(\Vert u\Vert _{X_{0}}^{p}+\Vert v\Vert _{X_{0}}^{p} \bigr)^{\frac{1}{p}} \\ =& \biggl( \int_{Q} \frac{|u(x)-u(y)|^{p}}{|x-y|^{n+sp}}\,dx\,dy+ \int_{Q} \frac{|v(x)-v(y)|^{p}}{|x-y|^{n+sp}}\,dx\,dy \biggr)^{\frac{1}{p}}. \end{aligned}
(1.7)

### Definition 1.1

We say that $$(u,v)\in E$$ is a weak solution of problem (1.1) if for any $$(\phi,\psi)\in E$$ one has

\begin{aligned}& M\bigl(\Vert u\Vert _{X_{0}}\bigr) \int_{Q}\frac {|u(x)-u(y)|^{p-2}(u(x)-u(y))(\phi(x)-\phi (y))}{|x-y|^{n+sp}}\,dx\,dy \\& \qquad{}+M\bigl(\Vert v\Vert _{X_{0}}\bigr) \int_{Q}\frac {|v(x)-v(y)|^{p-2}(v(x)-v(y))(\psi (x)-\psi (y))}{|x-y|^{n+sp}}\,dx\,dy \\& \quad= \int_{\Omega}\bigl(\lambda|u|^{q-2}u\phi+ \mu|v|^{q-2}v\psi\bigr)\, dx+\frac {\alpha }{\alpha+\beta} \int_{\Omega}|u|^{\alpha-2}u|v|^{\beta}\phi\, dx \\& \qquad{}+\frac{\beta}{\alpha+\beta} \int_{\Omega}|u|^{\alpha}|v|^{\beta -2}v\psi \,dx. \end{aligned}
(1.8)

The main result of this paper is as follows.

### Theorem 1.2

Let $$s\in(0,1)$$, $$n>sp$$. If $$1< q< p< p(\tau+1)<\alpha+\beta<p^{*}$$, then there exists $$\Lambda_{0}>0$$ such that for $$0<\lambda+\mu <\Lambda_{0}$$ problem (1.1) has at least two solutions.

### Remark 1

To our best knowledge, there is no similar result of this system for the case $$p=2$$.

This paper is organized as follows. In Section 2, we give some preliminaries of a Nehari manifold and a variational setting of problem (1.1). Section 3 gives the proof of Theorem 1.2.

## The variational setting

Define a functional $$I(u,v):E\rightarrow\mathbb{R}$$ as follows:

$$I(u,v)=\frac{k}{p}\bigl\Vert (u,v)\bigr\Vert ^{p}+\frac{l}{\sigma}\bigl\Vert (u,v)\bigr\Vert ^{\sigma}- \frac {1}{m} \int_{\Omega}|u|^{\alpha}|v|^{\beta}\,dx - \frac{1}{q}G(u,v),$$
(2.1)

where $$\sigma=p(\tau+1)$$, and $$m=\alpha+\beta$$, and

$$G(u,v)= \int_{\Omega}\bigl(\lambda|u|^{q}+\mu|v|^{q} \bigr)\,dx.$$

By a direct computation, we know that $$I(u,v)\in C^{1}(E,\mathbb{R})$$ and, for $$\forall(\phi,\psi)\in E$$, there holds

\begin{aligned} \bigl\langle I'(u,v),(\phi,\psi)\bigr\rangle = &M \bigl(\Vert u\Vert _{X_{0}}\bigr) \int_{Q}\frac {|u(x)-u(y)|^{p-2}(u(x)-u(y))(\phi(x)-\phi(y))}{|x-y|^{n+sp}}\,dx\, dy \\ &{}+M\bigl(\Vert v\Vert _{X_{0}}\bigr) \int_{Q}\frac {|v(x)-v(y)|^{p-2}(v(x)-v(y))(\psi (x)-\psi (y))}{|x-y|^{n+sp}}\,dx\,dy \\ &{}- \int_{\Omega}\bigl(\lambda|u|^{q-2}u\phi+ \mu|v|^{q-2}v\psi\bigr)\, dx-\frac {\alpha }{m} \int_{\Omega}|u|^{\alpha-2}u|v|^{\beta}\phi\,dx \\ &{}-\frac{\beta}{m} \int_{\Omega}|u|^{\alpha}|v|^{\beta-2}v\psi\,dx. \end{aligned}
(2.2)

Then the weak solutions of problem (1.1) correspond to the critical points of the functional I. Since I is not bounded below on E, we consider it on the Nehari manifold

$$N=\bigl\{ (u,v)\in E\setminus{(0,0)}|\bigl\langle I'(u,v),(u,v)\bigr\rangle =0\bigr\} .$$

From (2.2), we have

$$\bigl\langle I'(u,v),(u,v)\bigr\rangle = k\bigl\Vert (u,v)\bigr\Vert ^{p}+l\bigl\Vert (u,v)\bigr\Vert ^{\sigma}- \int _{\Omega}|u|^{\alpha}|v|^{\beta}\,dx -G(u,v).$$
(2.3)

Thus, $$(u,v)\in N$$ if and only if

$$k\bigl\Vert (u,v)\bigr\Vert ^{p}+l\bigl\Vert (u,v) \bigr\Vert ^{\sigma}- \int_{\Omega}|u|^{\alpha}|v|^{\beta}\,dx -G(u,v)=0.$$
(2.4)

Particularly, the following equality holds on N:

\begin{aligned} I(u,v) =&k\biggl(\frac{1}{p}-\frac{1}{q}\biggr)\bigl\Vert (u,v)\bigr\Vert ^{p}+l\biggl(\frac {1}{\sigma }- \frac {1}{q}\biggr)\bigl\Vert (u,v)\bigr\Vert ^{\sigma}-\biggl( \frac{1}{m}-\frac{1}{q}\biggr) \int _{\Omega}|u|^{\alpha }|v|^{\beta}\,dx \\ =&k\biggl(\frac{1}{p}-\frac{1}{m}\biggr)\bigl\Vert (u,v)\bigr\Vert ^{p}+l\biggl(\frac {1}{\sigma}-\frac {1}{m}\biggr)\bigl\Vert (u,v)\bigr\Vert ^{\sigma}-\biggl(\frac{1}{q}- \frac{1}{m}\biggr)G(u,v). \end{aligned}
(2.5)

Define

$$\Phi(u,v)=\bigl\langle I'(u,v),(u,v)\bigr\rangle ,\quad \forall(u,v) \in E.$$

Then, for any $$(u,v)\in N$$,

\begin{aligned}& \bigl\langle \Phi'(u,v),(u,v)\bigr\rangle \\& \quad=kp\bigl\Vert (u,v)\bigr\Vert ^{p}+l\sigma\bigl\Vert (u,v) \bigr\Vert ^{\sigma}-m \int_{\Omega}|u|^{\alpha}|v|^{\beta}\,dx -qG(u,v) \\& \quad=k(p-m)\bigl\Vert (u,v)\bigr\Vert ^{p}+l(\sigma-m)\bigl\Vert (u,v)\bigr\Vert ^{\sigma}-(q-m)G(u,v) \\& \quad=k(p-q)\bigl\Vert (u,v)\bigr\Vert ^{p}+l(\sigma-q)\bigl\Vert (u,v)\bigr\Vert ^{\sigma}-(m-q) \int_{\Omega}|u|^{\alpha}|v|^{\beta}\,dx. \end{aligned}
(2.6)

Thus, it is natural to split N into three parts:

\begin{aligned}& N^{+}=\bigl\{ (u,v)\in N:\bigl\langle \Phi'(u,v),(u,v) \bigr\rangle >0\bigr\} , \\& N^{-}=\bigl\{ (u,v)\in N:\bigl\langle \Phi'(u,v),(u,v)\bigr\rangle < 0 \bigr\} , \\& N^{0}=\bigl\{ (u,v)\in N:\bigl\langle \Phi'(u,v),(u,v)\bigr\rangle =0\bigr\} . \end{aligned}
(2.7)

We now derive some properties of $$N^{+}$$, $$N^{-}$$ and $$N^{0}$$.

### Lemma 2.1

I is coercive and bounded below on N.

### Proof

By Hölder’s inequality and (1.6), we have

\begin{aligned} \int_{\Omega}\lambda|u|^{q}\,dx \leq&\lambda \biggl( \int_{\Omega}1\, dx \biggr)^{\frac{m-q}{m}} \biggl( \int_{\Omega}|u|^{m}\,dx \biggr)^{\frac{q}{m}} = \lambda|\Omega|^{\frac{m-q}{m}}\|u\|_{m}^{q} \\ \leq&\lambda|\Omega |^{\frac {m-q}{m}}S^{-\frac{q}{p}}\|u\|_{X_{0}}^{q} \leq\lambda|\Omega|^{\frac{m-q}{m}}S^{-\frac{q}{p}}\bigl\Vert (u,v)\bigr\Vert ^{q}. \end{aligned}

Similarly,

$$\int_{\Omega}\mu|v|^{q}\,dx \leq\mu| \Omega|^{\frac{m-q}{m}}S^{-\frac{q}{p}}\Vert v\Vert _{X_{0}}^{q} \leq\mu|\Omega|^{\frac{m-q}{m}}S^{-\frac{q}{p}}\bigl\Vert (u,v)\bigr\Vert ^{q}.$$

Then

$$G(u,v)\leq(\lambda+\mu)|\Omega|^{\frac{m-q}{m}}S^{-\frac {q}{p}} \bigl\Vert (u,v)\bigr\Vert ^{q}.$$
(2.8)

It follows from (2.5) and (2.8) that

\begin{aligned} I(u,v) \geq& k\biggl(\frac{1}{p}-\frac{1}{m}\biggr) \bigl\Vert (u,v)\bigr\Vert ^{p}+l\biggl(\frac{1}{\sigma }- \frac{1}{m}\biggr)\bigl\Vert (u,v)\bigr\Vert ^{\sigma} \\ &{}-\biggl(\frac{1}{q}-\frac{1}{m}\biggr) (\lambda+\mu)| \Omega|^{\frac {m-q}{m}}S^{-\frac{q}{p}}\bigl\Vert (u,v)\bigr\Vert ^{q}. \end{aligned}
(2.9)

Since $$q< p\leq\sigma< m$$, from inequality (2.9), the functional I is coercive and bounded below on N. The proof is completed. □

### Lemma 2.2

There exists $$\Lambda_{0}>0$$, given by

$$\Lambda_{0}=\frac{k(m-p)}{(m-q)|\Omega|^{\frac{m-q}{m}}S^{-\frac{q}{p}}} \biggl(\frac{k(p-q)}{(m-q)S^{-\frac{m}{q}}} \biggr)^{\frac{p-q}{m-p}},$$

such that for any $$0<\lambda+\mu<\Lambda_{0}$$ we have $$N^{0}=\emptyset$$.

### Proof

We argue by contradiction. Assume that there exist $$\lambda ,\mu>0$$ with $$0<\lambda+\mu<\Lambda_{0}$$ such that $$N^{0}\neq\emptyset$$. Then, for $$(u,v)\in N^{0}$$, we have

$$\bigl\langle I'(u,v),(u,v)\bigr\rangle =0 \quad\mbox{and}\quad \bigl\langle \Phi '(u,v),(u,v)\bigr\rangle =0.$$

Then it follows from (2.5)-(2.8) that

$$\bigl\Vert (u,v)\bigr\Vert \leq \biggl(\frac{(m-q)(\lambda+\mu)|\Omega |^{\frac {m-q}{m}}S^{-\frac{q}{p}}}{k(m-p)} \biggr)^{\frac{1}{p-q}}.$$
(2.10)

On the other hand, by Young’s inequality, we have

\begin{aligned} \begin{aligned}[b] \int_{\Omega}|u|^{\alpha}|v|^{\beta}\,dx&\leq \frac{\alpha}{m} \int _{\Omega}|u|^{m} \,dx+\frac{\beta}{m} \int_{\Omega}|v|^{m} \,dx \\ & \leq\frac{\alpha}{m}S^{-\frac{m}{q}}\|u\|_{X_{0}}^{m}+ \frac{\beta }{m}S^{-\frac{m}{q}}\|v\|_{X_{0}}^{m} \leq S^{-\frac{m}{q}}\bigl\Vert (u,v)\bigr\Vert ^{m}. \end{aligned} \end{aligned}
(2.11)

From (2.5)-(2.7) and (2.11) it follows that

$$k(p-q)\bigl\Vert (u,v)\bigr\Vert ^{p}\leq(m-q)S^{-\frac{m}{q}} \bigl\Vert (u,v)\bigr\Vert ^{m}.$$

We have

$$\bigl\Vert (u,v)\bigr\Vert \geq \biggl(\frac{k(p-q)}{(m-q)S^{-\frac{m}{q}}} \biggr)^{\frac{1}{m-p}}.$$
(2.12)

By (2.10) and (2.12),

$$\lambda+\mu\geq\frac{k(m-p)}{(m-q)|\Omega|^{\frac {m-q}{m}}S^{-\frac{q}{p}}} \biggl(\frac{k(p-q)}{(m-q)S^{-\frac{m}{q}}} \biggr)^{\frac {p-q}{m-p}}=\Lambda_{0},$$

which contradicts $$0<\lambda+\mu<\Lambda_{0}$$. □

By Lemmas 2.1 and 2.2, we write $$N=N^{+}+N^{-}$$ for $$0<\lambda +\mu<\Lambda_{0}$$, and I is coercive and bounded from below on $$N^{+}$$ and $$N^{-}$$. We define

$$C^{+}=\inf_{(u,v)\in N^{+}} I(u,v) ,\qquad C^{-}=\inf_{(u,v)\in N^{-}} I(u,v).$$

As proved in , we have the following lemma.

### Lemma 2.3

For $$0<\lambda+\mu<\Lambda_{0}$$, suppose that $$(u_{0},v_{0})$$ is a local minimizer for I on N. Then, if $$(u_{0},v_{0})\notin N^{0}$$, $$(u_{0},v_{0})$$ is a critical point of I.

### Lemma 2.4

1. (a)

If $$0<\lambda+\mu<\Lambda_{0}$$, then $$C^{+}<0$$.

2. (b)

If $$0<\lambda+\mu<\frac{q}{p}\Lambda_{0}$$, then $$\exists d_{0}>0$$ such that $$C^{-}>d_{0}$$.

### Proof

(a) Let $$(u,v)\in N^{+}$$, it follows from (2.6) and (2.7) that

$$\int_{\Omega}|u|^{\alpha}|v|^{\beta}\,dx< \frac{k(p-q)}{m-q}\bigl\Vert (u,v)\bigr\Vert ^{p}+\frac{l(\sigma -q)}{m-q} \bigl\Vert (u,v)\bigr\Vert ^{\sigma}.$$
(2.13)

Put (2.13) into (2.5),

$$I(u,v)< -\frac{k(p-q)}{mpq}\bigl\Vert (u,v)\bigr\Vert ^{p}- \frac {l(p-q)(m-p)}{mpq}\bigl\Vert (u,v)\bigr\Vert ^{\sigma}< 0,$$

which implies $$C^{+}=\inf_{(u,v)\in N^{+}} I(u,v)<0$$.

(b) Let $$(u,v)\in N^{-}$$. By (2.5) and (2.8),

\begin{aligned} I(u,v) \geq&\frac{k(m-p)}{pm}\bigl\Vert (u,v)\bigr\Vert ^{p}-\frac {m-q}{mq}G(u,v) \\ \geq&\frac{k(m-p)}{pm}\bigl\Vert (u,v)\bigr\Vert ^{p}- \frac {m-q}{mq}(\lambda+\mu )|\Omega |^{\frac{m-q}{m}}S^{-\frac{q}{p}}\bigl\Vert (u,v)\bigr\Vert ^{q} \\ =&\bigl\Vert (u,v)\bigr\Vert ^{q} \biggl(\frac{k(m-p)}{pm}\bigl\Vert (u,v)\bigr\Vert ^{p-q}-\frac {m-q}{mq}(\lambda+\mu)| \Omega|^{\frac{m-q}{m}}S^{-\frac {q}{p}} \biggr). \end{aligned}
(2.14)

Combining (2.12) with (2.14), we have

$$I(u,v)\geq \biggl(\frac{k(p-q)}{(m-q)S^{-\frac{m}{q}}} \biggr)^{\frac{q}{m-p}} \biggl( \frac{k(m-p)}{pm} \biggl(\frac{k(p-q)}{(m-q)S^{-\frac {m}{q}}} \biggr) ^{\frac{p-q}{m-p}}- \frac{m-q}{mq}(\lambda+\mu)|\Omega|^{\frac {m-q}{m}}S^{-\frac{q}{p}} \biggr).$$

Clearly, if $$0<\lambda+\mu<\Lambda_{0}$$, then there exists $$d_{0}(p,q,\alpha,\beta,S)>0$$ such that $$C^{-}=\inf_{(u,v)\in N^{-}} I(u, v)>d_{0}$$. □

For each $$(u,v)\in E$$, let

$$\eta(t)=kt^{p-q}\bigl\Vert (u,v)\bigr\Vert ^{p}+lt^{\sigma-q}\Vert u,v\Vert ^{\sigma}-t^{m-q} \int _{\Omega}|u|^{\alpha}|v|^{\beta}\,dx.$$
(2.15)

Then

$$\eta'(t)=t^{p-q-1}E(t),$$

where

$$E(t)=k(p-q)\bigl\Vert (u,v)\bigr\Vert ^{p}+l(\sigma-q)t^{\sigma-p} \bigl\Vert (u,v)\bigr\Vert ^{\sigma}-(m-q)t^{m-p} \int_{\Omega}|u|^{\alpha}|v|^{\beta}\,dx.$$

Define

$$t^{*}= \biggl(\frac{l(\sigma-q)(\sigma-p)\|(u,v)\|^{\sigma}}{(m-q)(m-p)\int _{\Omega}|u|^{\alpha}|v|^{\beta}\,dx} \biggr)^{\frac{1}{m-\sigma}}.$$

It is easy to check that $$E(t)$$ increases for $$t\in[0,t^{*})$$ and decreases for $$t\in(t^{*},\infty)$$, $$E(t)$$ achieves its maximum at $$t^{*}$$. Since $$E(t)\rightarrow0$$ as $$t\rightarrow0^{+}$$ and $$E(t)\rightarrow-\infty$$ as $$t\rightarrow\infty$$ and there exists unique $$t_{l}$$, $$0< t^{*}< t_{l}$$, such that $$E(t_{l})=0$$, so $$\eta(t)$$ achieves its maximum at $$t_{l}$$, increasing for $$t\in[0,t_{l})$$ and decreasing for $$t\in(t_{l},\infty)$$. When $$l=0$$, we have

$$t_{0}= \biggl(\frac{k(p-q)\Vert (u,v)\Vert ^{p}}{(m-q)\int_{\Omega}|u|^{\alpha}|v|^{\beta}\,dx} \biggr)^{\frac{q}{m-p}}.$$
(2.16)

Obviously, $$E(t_{0})=E(t_{l})=0$$ and $$t_{0}\leq t_{l}$$ for $$l\geq0$$. Thus

$$\eta(t_{l})\geq\frac{k(m-p)}{m-q}t_{l}^{p-q} \bigl\Vert (u,v)\bigr\Vert ^{p}\geq \frac {k(m-p)}{m-q}t_{0}^{p-q} \bigl\Vert (u,v)\bigr\Vert ^{p}=\eta(t_{0}).$$
(2.17)

Set

\begin{aligned}& \Psi_{0}(t)=\Phi(tu,tv)=\bigl\langle I'(tu,tv) (tu,tv) \bigr\rangle \\& \hphantom{\Psi_{0}(t)}=kt^{p}\bigl\Vert (u,v)\bigr\Vert ^{p}+lt^{\sigma}\bigl\Vert (u,v)\bigr\Vert ^{\sigma}-t^{m} \int_{\Omega}|u|^{\alpha}|v|^{\beta}\,dx-t^{q}G(u,v), \\& \Psi_{1}(t)=\bigl\langle \Phi'(tu,tv),(tu,tv)\bigr\rangle \\& \hphantom{\Psi_{1}(t)}=kpt^{p}\bigl\Vert (u,v)\bigr\Vert ^{p}+l \sigma t^{\sigma}\bigl\Vert (u,v)\bigr\Vert ^{\sigma}-mt^{m} \int_{\Omega}|u|^{\alpha}|v|^{\beta}\,dx-qt^{q}G(u,v). \end{aligned}

Then

$$\Psi_{0}(t)=t^{q}\bigl(\eta(t)-G(u,v)\bigr).$$
(2.18)

### Lemma 2.5

$$(tu,tv)\in N^{+}$$ (or $$N^{-}$$) if and only if $$\Psi_{1}(t)>0$$ (or $$\Psi_{1}(t)<0$$).

### Proof

By (2.7), it is clear that $$(tu,tv)\in N^{+}$$ (or $$N^{-}$$) if and only if $$(tu,tv)\in N$$ and $$\langle\Phi'(tu,tv),(tu,tv)\rangle>0$$ (<0) for $$t>0$$. Note that

$$\Psi_{0}(t)=\Phi(tu,tv)=\bigl\langle I'(tu,tv), (tu,tv) \bigr\rangle ,\qquad \Psi _{1}(t)=\bigl\langle \Phi'(tu,tv),(tu,tv) \bigr\rangle .$$

Hence, $$(tu,tv)\in N^{+}$$ if and only if $$\Psi_{0}(t)=0$$ and $$\Psi_{1}(t)>0$$. □

### Lemma 2.6

For each $$(u,v)\in E\setminus{(0,0)}$$ and $$0<\lambda+\mu<\Lambda_{0}$$, there exist $$0< t_{1}< t_{l}< t_{2}$$ such that $$(t_{1}u,t_{1}v)\in N^{+}$$, $$(t_{2}u,t_{2}v)\in N^{-}$$, and

$$I(t_{1}u,t_{1}v)=\inf_{0\leq t\leq t_{l}}I(tu,tv),\qquad I(t_{2}u,t_{2}v)=\sup_{t\geq 0}I(tu,tv).$$

### Proof

Set

\begin{aligned} \Psi_{2}(t) =&I(tu,tv) \\ =&\frac{kt^{p}}{p}\bigl\Vert (u,v)\bigr\Vert ^{p}+ \frac{lt^{\sigma}}{\sigma}\bigl\Vert (u,v)\bigr\Vert ^{\sigma}- \frac{t^{m}}{m} \int_{\Omega}|u|^{\alpha}|v|^{\beta}\,dx- \frac{t^{q}}{q}G(u,v). \end{aligned}

Since $$0<\lambda+\mu<\Lambda_{0}$$, by (2.8), (2.15) and (2.17), we have

$$G(u,v)\leq(\lambda+\mu)|\Omega|^{\frac{m-q}{m}}S^{-\frac {q}{p}}\bigl\Vert (u,v) \bigr\Vert ^{q} \leq\eta(t_{0})\leq\eta(t_{l}).$$

Thus, there exist $$t_{1}$$ and $$t_{2}$$ such that $$0< t_{1}< t_{l}< t_{2}$$ and $$\eta(t_{1})=\eta(t_{2})=G(u,v)$$. It follows from (2.18) that $$\Psi_{0}(t_{1})=0$$ and $$\Psi_{0}(t_{2})=0$$, then $$(t_{1}u,t_{1}v)\in N$$ and $$(t_{2}u,t_{2}v)\in N$$. $$\Psi_{1}(t_{1})=(t_{1})^{q+1}\eta'(t_{1})>0$$. By Lemma 2.5, one has $$(t_{1}u,t_{1}v)\in N^{+}$$. Meanwhile, $$\Psi_{1}(t_{2})=(t_{2})^{q+1}\eta'(t_{2})<0$$, we obtain $$(t_{2}u,t_{2}v)\in N^{-}$$. By a direct calculation, we have $$\Psi_{2}'(t)=t^{q-1}(\eta(t)-G(u,v))$$. Since $$\Psi_{2}'(t)<0$$ for $$t\in[0,t_{1})$$ and $$\Psi_{2}'(t)>0$$ for $$t\in[t_{1},t_{l})$$, $$I(t_{1}u,t_{1}v)=\inf_{0\leq t\leq t_{l}}I(tu,tv)$$. Furthermore, we find that $$\Psi_{2}'(t)>0$$ for $$t\in[t_{1},t_{2})$$, $$\Psi _{2}'(t)<0$$ for $$t\in[t_{2},+\infty)$$ and $$\Psi_{2}(t)\leq0$$ for $$t\in[0,t_{1}]$$. Since $$(t_{2}u,t_{2}v)\in N^{-}$$, by Lemma 2.4, we obtain $$\Psi_{2}(t_{2})>0$$. Then $$I(t_{2}u,t_{2}v)=\sup_{t\geq0}I(tu,tv)$$. □

## Proof of the main result

### Lemma 3.1

If $$0<\lambda+\mu<\Lambda_{0}$$, then the functional I has a minimizer $$(u_{1},v_{1})$$ in $$N^{+}$$ satisfying

1. (i)

$$I(u_{1},v_{1})=C^{+}<0$$;

2. (ii)

$$(u_{1},v_{1})$$ is a solution of problem (1.1).

### Proof

Since I is bounded from below on $$N^{+}$$, there exists a minimizing sequence $$\{(u_{n}, v_{n})\}\in N^{+}$$ such that

$$\lim_{n\rightarrow\infty}I(u_{n},v_{n})=\inf _{(u,v)\in N^{+}}I(u,v)=C^{+}.$$

Since $$I(u,v)$$ is coercive and bounded from below on N, then $$\{ (u_{n},v_{n})\}$$ is bounded on E. Then there exists $$(u_{1},v_{1})\in E$$, up to a subsequence, that we still denote by $$\{(u_{n},v_{n})\}$$, such that, as $$n\rightarrow \infty$$,

\begin{aligned}& u_{n}\rightharpoonup u_{1},\qquad v_{n} \rightharpoonup v_{1},\quad \mbox{in }L^{r}(\Omega), \\& u_{n}(x)\rightarrow u_{1}(x),\qquad v_{n}(x) \rightarrow v_{1}(x),\quad \mbox{a.e. in }\Omega \end{aligned}

for any $$1\leq r< p^{*}$$, and by , Theorem IV-9, there exists $$l(x)\in L^{r}(\mathbb{R}^{n})$$ such that

$$\bigl\vert u_{n}(x)\bigr\vert \leq l(x),\qquad \bigl\vert v_{n}(x)\bigr\vert \leq l(x),\quad \mbox{a.e. in } \mathbb{R}^{n}$$

for any $$1\leq r< p^{*}$$. By the dominated convergence theorem,

\begin{aligned} \lim_{n\rightarrow\infty} \int_{\Omega}\bigl(\lambda|u_{n}|^{q}+ \mu|v_{n}|^{q}\bigr)\,dx =& \int_{\Omega}\lim_{n\rightarrow\infty}\bigl( \lambda|u_{n}|^{q}+\mu |v_{n}|^{q}\bigr) \,dx \\ =& \int_{\Omega}\bigl(\lambda|u_{1}|^{q}+ \mu|v_{1}|^{q}\bigr)\,dx, \end{aligned}

and

$$\lim_{n\rightarrow\infty} \int_{\Omega}|u_{n}|^{\alpha}|v_{n}|^{\beta}\, dx= \int _{\Omega}|u_{1}|^{\alpha}|v_{1}|^{\beta}\,dx.$$

By Lemma 2.6, there exists $$t_{1}< t_{l}$$ such that $$(t_{1}u_{1},t_{1}v_{1})\in N^{+}$$ and $$\Psi_{0}(t_{1})=\langle I'(t_{1}u_{1},t_{1}v_{1}), (t_{1}u_{1},t_{1}v_{1})\rangle=0$$.

Next we show that $$(u_{n},v_{n})\rightarrow(u_{1},v_{1})$$ strongly in E. Suppose otherwise, then

$$\bigl\Vert (u_{1},v_{1})\bigr\Vert < \liminf _{n\rightarrow\infty}\bigl\Vert (u_{n},v_{n})\bigr\Vert .$$

As

\begin{aligned} \bigl\langle I'(t_{1}u_{n},t_{1}v_{n}),(t_{1}u_{n},t_{1}v_{n}) \bigr\rangle =&kt_{1}^{p}\bigl\Vert (u_{n},v_{n}) \bigr\Vert ^{p} +lt_{1}^{\sigma}\bigl\Vert (u_{n},v_{n})\bigr\Vert ^{\sigma}\\ &{}-t_{1}^{m} \int_{\Omega}|u_{n}|^{\alpha}|v_{n}|^{\beta}\,dx -t_{1}^{q}G(u_{n},v_{n}), \end{aligned}

and

\begin{aligned} \bigl\langle I'(t_{1}u_{1},t_{1}v_{1}),(t_{1}u_{1},t_{1}v_{1}) \bigr\rangle =&kt_{1}^{p}\bigl\Vert (u_{1},v_{1}) \bigr\Vert ^{p} +lt_{1}^{\sigma}\bigl\Vert (u_{1},v_{1})\bigr\Vert ^{\sigma}\\ &{}-t_{1}^{m} \int_{\Omega}|u_{1}|^{\alpha}|v_{1}|^{\beta}\,dx -t_{1}^{q}G(u_{1},v_{1}), \end{aligned}

we have

$$\lim_{n\rightarrow\infty}\bigl\langle I'(t_{1}u_{n},t_{1}v_{n}),(t_{1}u_{n},t_{1}v_{n}) \bigr\rangle > \bigl\langle I'(t_{1}u_{1},t_{1}v_{1}),(t_{1}u_{1},t_{1}v_{1}) \bigr\rangle =\Psi_{0}(t_{1})=0.$$

That is, $$\langle I'(t_{1}u_{n},t_{1}v_{n}),(t_{1}u_{n},t_{1}v_{n})\rangle>0$$ for n large enough. Since $$\{(u_{n},v_{n})\}\in N^{+}$$, it is easy to see that $$\langle I'(u_{n},v_{n}),(u_{n},v_{n})\rangle=0$$, and $$\langle I'(tu_{n},tv_{n}),(tu_{n},tv_{n})\rangle<0$$ for $$0< t<1$$. So we have $$t_{1}>1$$. On the other hand, $$I(tu_{1},tv_{1})$$ is decreasing on $$(0,t_{1})$$, So

$$I(t_{1}u_{1},t_{1}v_{1})\leq I(u_{1},v_{1})< \liminf_{n\rightarrow\infty }I(u_{n},v_{n})=C^{+}= \inf_{(u,v)\in N^{+}}I(u,v),$$

which is a contradiction. Hence $$(u_{n},v_{n})\rightarrow(u_{1},v_{1})$$ strongly in E. This implies

$$I(u_{n},v_{n})\rightarrow I(u_{1},v_{1})= \inf_{(u,v)\in N^{+}}I(u,v)=C^{+}\quad \mbox{as } n\rightarrow\infty.$$

Namely, $$(u_{1},v_{1})$$ is a minimizer of I on $$N^{+}$$, by Lemma 2.2, $$(u_{1},v_{1})$$ is a solution of problem (1.1). □

### Lemma 3.2

If $$0<\lambda+\mu<\Lambda_{0}$$, then the functional I has a minimizer $$(u_{2},v_{2})$$ in $$N^{-}$$ such that

1. (i)

$$I(u_{2},v_{2})=C^{-}$$;

2. (ii)

$$(u_{2},v_{2})$$ is a solution of problem (1.1).

### Proof

Since I is bounded from below on $$N^{-}$$, there exists a minimizing sequence $$\{(\bar{u}_{n}, \bar{v}_{n})\}\in N^{-}$$ such that

$$\lim_{n\rightarrow\infty}I(\bar{u}_{n},\bar{v}_{n})=C^{-}.$$

Since $$I(u,v)$$ is coercive, $$\{(\bar{u}_{n},\bar{v}_{n})\}$$ is bounded on E, up to a subsequence, we still denote it by $$\{(\bar{u}_{n},\bar{v}_{n})\}$$, then there exists $$(u_{2},v_{2})\in E$$ such that

$$\bar{u}_{n}\rightharpoonup u_{2}, \qquad \bar{v}_{n}\rightharpoonup v_{2},\quad \mbox{in } L^{r}(\Omega)$$

for any $$1\leq r< p^{*}$$, and by , Theorem IV-9, and the dominated convergence theorem,

$$\lim_{n\rightarrow\infty} G(\bar{u}_{n},\bar{v}_{n})= G(u_{2},v_{2}),$$

and

$$\lim_{n\rightarrow\infty} \int_{\Omega} \vert \bar{u}_{n}\vert ^{\alpha} \vert \bar {v}_{n}\vert ^{\beta}\,dx= \int_{\Omega} \vert u_{2}\vert ^{\alpha} \vert v_{2}\vert ^{\beta}\,dx.$$

By Lemma 2.6, there exists unique $$t_{2}$$ such that $$(t_{2}u_{2},t_{2}v_{2})\in N^{-}$$. Next we show that $$(\bar{u}_{n},\bar{v}_{n})\rightarrow(u_{2},v_{2})$$ strongly in E. The proof of this claim is by contradiction. If the claim were not true, then

$$\bigl\Vert (u_{2},v_{2})\bigr\Vert < \liminf _{n\rightarrow\infty}\bigl\Vert (\bar {u}_{n},\bar{v}_{n}) \bigr\Vert .$$

Since $$(\bar{u}_{n},\bar{v}_{n})\in N^{-}$$ and $$I(\bar{u}_{n},\bar {v}_{n})\geq I(t\bar{u}_{n}, t\bar{v}_{n})$$ for all $$t>0$$, then we have

$$I(t_{2}u_{2},t_{2}v_{2})< \liminf _{n\rightarrow\infty}I(t_{2}\bar{u}_{n},t_{2} \bar{v}_{n}) \leq\liminf_{n\rightarrow\infty}I(\bar{u}_{n}, \bar{v}_{n})=C^{-},$$

which is a contradiction. This implies

$$I(\bar{u}_{n},\bar{v}_{n})\rightarrow I(u_{2},v_{2})= \inf_{(u,v)\in N^{-}}I(u,v)=C^{-} \quad\mbox{as }n\rightarrow\infty.$$

Namely, $$(u_{2},v_{2})$$ is a minimizer of I on $$N^{-}$$, by Lemma 2.2, $$(u_{2},v_{2})$$ is a solution of problem (1.1). □

### Proof of Theorem 1.2

By Lemmas 3.1 and 3.2, we have that for $$0<\lambda+\mu<\Lambda_{0}$$, problem (1.1) has two solutions $$(u_{1},v_{1})\in N^{+}$$ and $$(u_{2},v_{2})\in N^{-}$$ in E. Since $$N^{+}\cap N^{-}=\emptyset$$, then these two solutions are distinct. □

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## Acknowledgements

The authors thank the anonymous referees for invaluable comments and insightful suggestions.

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