Existence of three solutions for a nonlocal elliptic system of $(p,q)$-Kirchhoff type

Abstract

In this paper, we study the solutions of a nonlocal elliptic system of $(p,q)$-Kirchhoff type on a bounded domain based on the three critical points theorem introduced by Ricceri. Firstly, we establish the existence of three weak solutions under appropriate hypotheses; then, we prove the existence of at least three weak solutions for the nonlocal elliptic system of $(p,q)$-Kirchhoff type.

1 Introduction and main results

We consider the boundary problem involving $(p,q)$-Kirchhoff

$$\{\begin{array}{c}-{[M_1({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^p)]}^{p-1}{\mathrm{\Delta }}_{p}u=\lambda F_u(x,u,\upsilon )+\mu G_u(x,u,\upsilon ),\text{in }\mathrm{\Omega },\hfill \\ -{[M_2({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }\upsilon {|}^q)]}^{q-1}{\mathrm{\Delta }}_q\upsilon =\lambda F_{\upsilon }(x,u,\upsilon )+\mu G_v(x,u,\upsilon ),\text{in }\mathrm{\Omega },\hfill \\ u=\upsilon =0,\text{on }\partial \mathrm{\Omega },\hfill \end{array}$$

(1.1)

where $\mathrm{\Omega }\subset R^N$ ($N\ge 1$) is a bounded smooth domain, $\lambda ,\mu \in [0,+\mathrm{\infty })$, $p>N$, $q>N$, ${\mathrm{\Delta }}_p$ is the p-Laplacian operator ${\mathrm{\Delta }}_{p}u=div(|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u)$. $F,G:\mathrm{\Omega }\times \mathrm{R}\times \mathrm{R}\mapsto \mathrm{R}$ are functions such that $F(\cdot ,s,t)$, $G(\cdot ,s,t)$ are measurable in Ω for all $(s,t)\in \mathrm{R}\times \mathrm{R}$ and $F(x,\cdot ,\cdot )$, $G(x,\cdot ,\cdot )$ are continuously differentiable in $\mathrm{R}\times \mathrm{R}$ for a.e. $x\in \mathrm{\Omega }$. $F_i$ is the partial derivative of F with respect to i, $i=u,v$, so is $G_i$. $M_i:R^{+}\to R$, $i=1,2$, are continuous functions which satisfy the following bounded conditions.

• (M) There exist two positive constants $m_0$, $m_1$ such that

  $$m_0\le M_i(t)\le m_1,\mathrm{\forall }t\ge 0,i=1,2.$$

  (1.2)

Here and in the sequel, X denotes the Cartesian product of two Sobolev spaces $W_0^{1,p}(\mathrm{\Omega })$ and $W_0^{1,q}(\mathrm{\Omega })$, i.e., $X=W_0^{1,p}(\mathrm{\Omega })\times W_0^{1,q}(\mathrm{\Omega })$. The reflexive real Banach space X is endowed with the norm

$$\parallel (u,\upsilon )\parallel ={\parallel u\parallel }_p+{\parallel \upsilon \parallel }_q,{\parallel u\parallel }_p={({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^p)}^{1/p},{\parallel \upsilon \parallel }_q={({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }\upsilon {|}^q)}^{1/q}.$$

Since $p>N$ and $q>N$, $W_0^{1,p}(\mathrm{\Omega })$ and $W_0^{1,q}(\mathrm{\Omega })$ are compactly embedded in $C^0(\overline{\mathrm{\Omega }})$. Let

$$C=max\{\underset{u\in W_0^{1,p}(\mathrm{\Omega })\mathrm{\setminus }\{0\}}{sup}\frac{{max}_{x\in \overline{\mathrm{\Omega }}}\{|u(x){|}^p\}}{{\parallel u\parallel }_p^p},\underset{v\in W_0^{1,q}(\mathrm{\Omega })\mathrm{\setminus }\{0\}}{sup}\frac{{max}_{x\in \overline{\mathrm{\Omega }}}\{|\upsilon (x){|}^q\}}{{\parallel \upsilon \parallel }_q^q}\},$$

(1.3)

then one has $C<+\mathrm{\infty }$. Furthermore, it is known from [1] that

$$\underset{u\in W_0^{1,p}(\mathrm{\Omega })\mathrm{\setminus }\{0\}}{sup}\frac{{max}_{x\in \overline{\mathrm{\Omega }}}\{|u(x){|}^p\}}{{\parallel u\parallel }_p}\le \frac{N^{-1/p}}{\sqrt{\pi }}{\left(\mathrm{\Gamma }(1+\frac{N}{2})\right)}^{1/N}{\left(\frac{p-1}{p-N}\right)}^{1-1/p}|\mathrm{\Omega }{|}^{(1/N)-(1/p)},$$

where Γ is the gamma function and $|\mathrm{\Omega }|$ is the Lebesgue measure of Ω. As usual, by a weak solution of system (1.1), we mean any $(u,\upsilon )\in X$ such that

$$\begin{array}{r}{\left[M_1({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^p)\right]}^{p-1}{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u\mathrm{\nabla }\varphi +{\left[M_2({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }\upsilon {|}^q)\right]}^{q-1}{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }\upsilon {|}^{q-2}\mathrm{\nabla }\upsilon \mathrm{\nabla }\psi \\ -\lambda {\int }_{\mathrm{\Omega }}(F_u\varphi +F_v\psi )dx-\mu {\int }_{\mathrm{\Omega }}(G_u\varphi +G_v\psi )dx=0\end{array}$$

(1.4)

for all $(\varphi ,\psi )\in X$.

System (1.1) is related to the stationary version of a model established by Kirchhoff [2]. More precisely, Kirchhoff proposed the following model:

$$\rho \frac{{\partial }^{2}u}{\partial t^2}-(\frac{P_0}{h}+\frac{E}{2L}{\int }_0^L|\frac{\partial u}{\partial x}{|}^{2}dx)\frac{{\partial }^{2}u}{\partial x^2}=0,$$

(1.5)

which extends D’Alembert’s wave equation with free vibrations of elastic strings, where ρ denotes the mass density, $P_0$ denotes the initial tension, h denotes the area of the cross-section, E denotes the Young modulus of the material, and L denotes the length of the string. Kirchhoff’s model considers the changes in length of the string produced during the vibrations.

Later, (1.1) was developed into the following form:

$$u_{tt}-M({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^2)\mathrm{\Delta }u=f(x,u)\text{in }\mathrm{\Omega },$$

(1.6)

where $M:R^{+}\to R$ is a given function. After that, many authors studied the following problem:

$$-M({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^2)\mathrm{\Delta }u=f(x,u)\text{in }\mathrm{\Omega },u=0\text{on }\partial \mathrm{\Omega },$$

(1.7)

which is the stationary counterpart of (1.6). By applying variational methods and other techniques, many results of (1.7) were obtained, the reader is referred to [3–13] and the references therein. In particular, Alves et al. [[3], Theorem 4] supposed that M satisfies bounded condition (M) and $f(x,t)$ satisfies the condition

$$0<\upsilon F(x,t)\le f(x,t)t,\mathrm{\forall }|t|\ge R,x\in \mathrm{\Omega }\text{ for some }v>2\text{ and }R>0,$$

(AR)

where $F(x,t)={\int }_0^{t}f(x,s)ds$; one positive solution for (1.7) was given.

In [14], using Ekeland’s variational principle, Corrêa and Nascimento proved the existence of a weak solution for the boundary problem associated with the nonlocal elliptic system of p-Kirchhoff type

$$\{\begin{array}{c}-{[M_1({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^p)]}^{p-1}{\mathrm{\Delta }}_{p}u=f(u,\upsilon )+{\rho }_1(x),\text{in }\mathrm{\Omega },\hfill \\ -{[M_2({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }\upsilon {|}^p)]}^{p-1}{\mathrm{\Delta }}_p\upsilon =g(,u,\upsilon )+{\rho }_2(x),\text{in }\mathrm{\Omega },\hfill \\ \frac{\partial u}{\partial \eta }=\frac{\partial \upsilon }{\partial \eta }=0,\text{on }\partial \mathrm{\Omega },\hfill \end{array}$$

(1.8)

where η is the unit exterior vector on ∂ Ω, and $M_i$, ${\rho }_i$ ($i=1,2$), f, g satisfy suitable assumptions.

In [15], when $\mu =0$ in (1.1), Bitao Cheng et al. studied the existence of two solutions and three solutions of the following nonlocal elliptic system:

$$\{\begin{array}{c}-{[M_1({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^p)]}^{p-1}{\mathrm{\Delta }}_{p}u=\lambda F_u(x,u,\upsilon ),\text{in }\mathrm{\Omega },\hfill \\ -{[M_2({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }\upsilon {|}^q)]}^{q-1}{\mathrm{\Delta }}_q\upsilon =\lambda F_{\upsilon }(x,u,\upsilon ),\text{in }\mathrm{\Omega },\hfill \\ u=\upsilon =0,\text{on }\partial \mathrm{\Omega }.\hfill \end{array}$$

(1.9)

In this paper, our objective is to prove the existence of three solutions of problem (1.1) by applying the three critical points theorem established by Ricceri [16]. Our result, under appropriate assumptions, ensures the existence of an open interval $\mathrm{\Lambda }\subset [0,+\mathrm{\infty })$ and a positive real number ρ such that, for each $\lambda \in \mathrm{\Lambda }$, problem (1.1) admits at least three weak solutions whose norms in X are less than ρ. The purpose of the present paper is to generalize the main result of [15].

Now, for every $x_0\in \mathrm{\Omega }$ and choosing $R_1$, $R_2$ with $R_2>R_1>0$, such that $B(x_0,R_2)\subseteq \mathrm{\Omega }$, where $B(x,R)=\{y\in R^N:|y-x|<R\}$, put

$${\alpha }_1={\alpha }_1(N,p,R_1,R_2)=\frac{C^{1/p}{(R_2^N-R_1^N)}^{1/p}}{R_2-R_1}{\left(\frac{{\pi }^{N/2}}{\mathrm{\Gamma }(1+N/2)}\right)}^{1/p},$$

(1.10)

$${\alpha }_2={\alpha }_2(N,q,R_1,R_2)=\frac{C^{1/q}{(R_2^N-R_1^N)}^{1/q}}{R_2-R_1}{\left(\frac{{\pi }^{N/2}}{\mathrm{\Gamma }(1+N/2)}\right)}^{1/q}.$$

(1.11)

Moreover, let a, c be positive constants and define

$$\begin{array}{c}y(x)=\frac{a}{R_2-R_1}(R_2-{\left\{\sum _{i=1}^N{(x^i-x_0^i)}^2\right\}}^{1/2}),\mathrm{\forall }x\in B(x_0,R_2)\mathrm{\setminus }B(x_0,R_1),\hfill \\ A(c)=\{(s,t)\in R\times R:|s{|}^p+|t{|}^q\le c\},\hfill \\ M^{+}=max\{\frac{m_1^{p-1}}{p},\frac{m_1^{q-1}}{q}\},M_{-}=min\{\frac{m_0^{p-1}}{p},\frac{m_0^{q-1}}{q}\}.\hfill \end{array}$$

Our main result is stated as follows.

Theorem 1.1 Assume that $R_2>R_1>0$ such that $B(x_0,R_2)\subseteq \mathrm{\Omega }$, and suppose that there exist four positive constants a, b, γ and β with $\gamma <p$, $\beta <q$, ${(a{\alpha }_1)}^p+{(a{\alpha }_2)}^q>b{M}^{+}/M_{-}$, and a function $\alpha (x)\in L^{\mathrm{\infty }}(\mathrm{\Omega })$ such that

• (j1) $F(x,s,t)\ge 0$ for a.e. $x\in \mathrm{\Omega }\mathrm{\setminus }B(x_0,R_1)$ and all $(s,t)\in [0,a]\times [0,a]$;

• (j2) $[{(a{\alpha }_1)}^p+{(a{\alpha }_2)}^q]|\mathrm{\Omega }|{sup}_{(x,s,t)\in \mathrm{\Omega }\times A(b{M}^{+}/M_{-})}F(x,s,t)<b{\int }_{B(x_0,R_1)}F(x,a,a)dx$;

• (j3) $F(x,s,t)\le \alpha (x)(1+|s{|}^{\gamma }+|t{|}^{\beta })$ for a.e. $x\in \mathrm{\Omega }$ and all $(s,t)\in R\times R$;

• (j4) $F(x,0,0)=0$ for a.e. $x\in \mathrm{\Omega }$.

Then there exist an open interval $\mathrm{\Lambda }\subseteq [0,\mathrm{\infty })$ and a positive real number ρ with the following property: for each $\lambda \in \mathrm{\Lambda }$ and for two Carathéodory functions $G_u,G_v:\mathrm{\Omega }\times R\times R\mapsto R$ satisfying

• (j5) ${sup}_{\{|s|\le \xi ,|t|\le \xi \}}(|G_u(\cdot ,s,t)|+|G_v(\cdot ,s,t)|)\in L^1(\mathrm{\Omega })$ for all $\xi >0$,

there exists $\delta >0$ such that, for each $\mu \in [0,\delta ]$, problem (1.1) has at least three weak solutions $w_i=(u_i,{\upsilon }_i)\in X$ ($i=1,2,3$) whose norms $\parallel w_i\parallel$ are less than ρ.

2 Proof of the main result

First we recall the modified form of Ricceri’s three critical points theorem (Theorem 1 in [16]) and Proposition 3.1 of [17], which is our primary tool in proving our main result.

Theorem 2.1 ([16], Theorem 1)

Suppose that X is a reflexive real Banach space and that $\mathrm{\Phi }:X\mapsto R$ is a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on $X^{\ast }$, and that Φ is bounded on each bounded subset of X; $\mathrm{\Psi }:X\mapsto R$ is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact; $I\subseteq R$ is an interval. Suppose that

$$\underset{\parallel x\parallel \to +\mathrm{\infty }}{lim}(\mathrm{\Phi }(x)+\lambda \mathrm{\Psi }(x))=+\mathrm{\infty }$$

for all $\lambda \in I$, and that there exists $h\in R$ such that

$$\underset{\lambda \in I}{sup}\underset{x\in X}{inf}(\mathrm{\Phi }(x)+\lambda (\mathrm{\Psi }(x)+h))<\underset{x\in X}{inf}\underset{\lambda \in I}{sup}(\mathrm{\Phi }(x)+\lambda (\mathrm{\Psi }(x)+h)).$$

(2.1)

Then there exist an open interval $\mathrm{\Lambda }\subseteq I$ and a positive real number ρ with the following property: for every $\lambda \in \mathrm{\Lambda }$ and every $C^1$ functional $J:X\mapsto R$ with compact derivative, there exists $\delta >0$ such that, for each $\mu \in [0,\delta ]$, the equation

$${\mathrm{\Phi }}^{\mathrm{\prime }}(x)+\lambda {\mathrm{\Psi }}^{\mathrm{\prime }}(x)+\mu J^{\mathrm{\prime }}(x)=0$$

has at least three solutions in X whose norms are less than ρ.

Proposition 2.1 ([17], Proposition 3.1)

Assume that X is a nonempty set and Φ, Ψ are two real functions on X. Suppose that there are $r>0$ and $x_0,x_1\in X$ such that

$$\mathrm{\Phi }(x_0)=-\mathrm{\Psi }(x_0)=0,\mathrm{\Phi }(x_1)>1,\underset{x\in {\mathrm{\Phi }}^{-1}([-\mathrm{\infty },r])}{sup}-\mathrm{\Psi }(x)<r\frac{-\mathrm{\Psi }(x_1)}{\mathrm{\Phi }(x_1)}.$$

Then, for each h satisfying

$$\underset{x\in {\mathrm{\Phi }}^{-1}([-\mathrm{\infty },r])}{sup}-\mathrm{\Psi }(x)<h<r\frac{-\mathrm{\Psi }(x_1)}{\mathrm{\Phi }(x_1)},$$

one has

$$\underset{\lambda \ge 0}{sup}\underset{x\in X}{inf}(\mathrm{\Phi }(x)+\lambda (\mathrm{\Psi }(x)+h))<\underset{x\in X}{inf}\underset{\lambda \ge 0}{sup}(\mathrm{\Phi }(x)+\lambda (\mathrm{\Psi }(x)+h)).$$

Before proving Theorem 1.1, we define a functional and give a lemma.

The functional $H:X\to R$ is defined by

$$\begin{array}{rl}H(u,v)=& \mathrm{\Phi }(u,v)+\lambda J(u,v)+\mu \psi (u,v)\\ =& \frac{1}{p}{\hat{M}}_1({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^p)+\frac{1}{q}{\hat{M}}_2({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }\upsilon {|}^q)\\ -\lambda {\int }_{\mathrm{\Omega }}F(x,u,v)dx-\mu {\int }_{\mathrm{\Omega }}G(x,u,v)dx\end{array}$$

(2.2)

for all $(u,\upsilon )\in X$, where

$${\hat{M}}_1={\int }_0^t{[M_1(s)]}^{p-1}ds,{\hat{M}}_2={\int }_0^t{[M_2(s)]}^{q-1}ds.$$

(2.3)

By conditions (M) and (j3), it is clear that $H\in C^1(X,R)$ and a critical point of H corresponds to a weak solution of system (1.1).

Lemma 2.2 Assume that there exist two positive constants a, b with ${(a{\alpha }_1)}^p+{(a{\alpha }_2)}^q>b{M}^{+}/M_{-}$ such that

• (j1) $F(x,s,t)\ge 0$, for a.e. $x\in \mathrm{\Omega }\mathrm{\setminus }B(x_0,R_1)$ and all $(s,t)\in [0,a]\times [0,a]$;

• (j2) $[{(a{\alpha }_1)}^p+{(a{\alpha }_2)}^q]|\mathrm{\Omega }|{sup}_{(x,s,t)\in \mathrm{\Omega }\times A(b{M}^{+}/M_{-})}F(x,s,t)<b{\int }_{B(x_0,R_1)}F(x,a,a)dx$.

Then there exist $r>0$ and $u_0\in W_0^{1,p}(\mathrm{\Omega })$, ${\upsilon }_0\in W_0^{1,q}(\mathrm{\Omega })$ such that

$$\mathrm{\Phi }(u_0,v_0)>r$$

and

$$|\mathrm{\Omega }|\underset{(x,s,t)\in \mathrm{\Omega }\times A(b{M}^{+}/M_{-})}{sup}F(x,s,t)\le \frac{b{M}^{+}}{C}\frac{{\int }_{\mathrm{\Omega }}F(x,u_0,{\upsilon }_0)dx}{\mathrm{\Phi }(u_0,{\upsilon }_0)}.$$

Proof We put

$$w_0(x)=\{\begin{array}{cc}0,\hfill & x\in \overline{\mathrm{\Omega }}\mathrm{\setminus }B(x_0,R_2),\hfill \\ \frac{a}{R_2-R_1}(R_2-{\{{\sum }_{i=1}^N(x^i-x_0^i)\}}^{1/2}),\hfill & x\in B(x_0,R_2)\mathrm{\setminus }B(x_0,R_1),\hfill \\ a,\hfill & x\in B(x_0,R_1),\hfill \end{array}$$

and $u_0(x)={\upsilon }_0(x)=w_0(x)$. Then we can verify easily $(u_0,{\upsilon }_0)\in X$ and, in particular, we have

$${\parallel u_0\parallel }_p^p=(R_2^N-R_1^N)\frac{{\pi }^{N/2}}{\mathrm{\Gamma }(1+N/2)}{\left(\frac{a}{R_2-R_1}\right)}^p,$$

(2.4)

and

$${\parallel {\upsilon }_0\parallel }_q^q=(R_2^N-R_1^N)\frac{{\pi }^{N/2}}{\mathrm{\Gamma }(1+N/2)}{\left(\frac{a}{R_2-R_1}\right)}^q.$$

(2.5)

Hence, we obtain from (1.10), (1.11), (2.4) and (2.5) that

$${\parallel u_0\parallel }_p^p={\parallel w_0\parallel }_p^p=\frac{{(a{\alpha }_1)}^p}{C},{\parallel {\upsilon }_0\parallel }_q^q={\parallel w_0\parallel }_q^q=\frac{{(a{\alpha }_2)}^q}{C}.$$

(2.6)

Under condition (M), by a simple computation, we have

$$M_{-}({\parallel u\parallel }_p^p+{\parallel \upsilon \parallel }_q^q)\le \mathrm{\Phi }(u,\upsilon )\le M^{+}({\parallel u\parallel }_p^p+{\parallel \upsilon \parallel }_q^q).$$

(2.7)

Setting $r=\frac{b{M}^{+}}{C}$ and applying the assumption of Lemma 2.2

$${(a{\alpha }_1)}^p+{(a{\alpha }_2)}^q>b{M}^{+}/M_{-},$$

from (2.6) and (2.7), we obtain

$$\mathrm{\Phi }(u_0,v_0)\ge M_{-}({\parallel u_0\parallel }_p^p+{\parallel v_0\parallel }_q^q)=\frac{M_{-}}{C}[{(a{\alpha }_1)}^p+{(a{\alpha }_2)}^q]>\frac{M_{-}}{C}\frac{b{M}^{+}}{M_{-}}=r.$$

Since, $0\le u_0\le a$, $0\le v_0\le a$ for each $x\in \mathrm{\Omega }$, from condition (j1) of Lemma 2.2, we have

$${\int }_{\mathrm{\Omega }\mathrm{\setminus }B(x_0,R_2)}F(x,u_0,{\upsilon }_0)dx+{\int }_{B(x_0,R_2)\mathrm{\setminus }B(x_0,R_1)}F(x,u_0,{\upsilon }_0)dx\ge 0.$$

Hence, based on condition (j2), we get

$$\begin{array}{rl}|\mathrm{\Omega }|\underset{(x,s,t)\in \mathrm{\Omega }\times A(b{M}^{+}/M_{-})}{sup}F(x,s,t)& <\frac{b}{{(a{\alpha }_1)}^p+{(a{\alpha }_2)}^q}{\int }_{B(x_0,R_1)}F(x,a,a)dx\\ =\frac{b{M}^{+}}{C}\frac{{\int }_{B(x_0,R_1)}F(x,a,a)dx}{M^{+}({(a{\alpha }_1)}^p+{(a{\alpha }_2)}^q)/C}\\ \le \frac{b{M}^{+}}{C}\frac{{\int }_{\mathrm{\Omega }\mathrm{\setminus }B(x_0,R_1)}F(x,u_0,{\upsilon }_0)dx+{\int }_{B(x_0,R_1)}F(x,u_0,{\upsilon }_0)dx}{M^{+}({\parallel u_0\parallel }_p^p+{\parallel {\upsilon }_0\parallel }_q^q)}\\ \le \frac{b{M}^{+}}{C}\frac{{\int }_{\mathrm{\Omega }}F(x,u_0,{\upsilon }_0)dx}{\mathrm{\Psi }(u_0,{\upsilon }_0)}.\end{array}$$

 □

Now, we can prove our main result.

Proof of Theorem 1.1 For each $(u,v)\in X$, let

$$\begin{array}{c}\mathrm{\Phi }(u,v)=\frac{{\hat{M}}_1({\parallel u\parallel }_p^p)}{p}+\frac{{\hat{M}}_2({\parallel v\parallel }_q^q)}{q},\hfill \\ \mathrm{\Psi }(u,\upsilon )=-{\int }_{\mathrm{\Omega }}F(x,u,\upsilon )dx,J(u,v)=-{\int }_{\mathrm{\Omega }}G(x,u,v)dx.\hfill \end{array}$$

From the assumption of Theorem 1.1, we know that Φ is a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional. Additionally, the Gâteaux derivative of Φ has a continuous inverse on $X^{\ast }$. Since $p>N$, $q>N$, Ψ and J are continuously Gâteaux differential functionals whose Gâteaux derivatives are compact. Obviously, Φ is bounded on each bounded subset of X. In particular, for each $(u,v),(\xi ,\eta )\in X$,

$$\begin{array}{c}\begin{array}{rl}〈{\mathrm{\Phi }}^{\mathrm{\prime }}(u,v),(\xi ,\eta )〉=& {\left[M_1({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^p)\right]}^{p-1}{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u\mathrm{\nabla }\xi \\ +{\left[M_2({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }\upsilon {|}^q)\right]}^{q-1}{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }\upsilon {|}^{q-2}\mathrm{\nabla }\upsilon \mathrm{\nabla }\eta ,\end{array}\hfill \\ 〈{\mathrm{\Psi }}^{\mathrm{\prime }}(u,v),(\xi ,\eta )〉=-{\int }_{\mathrm{\Omega }}{F}_u(x,u,v)\xi dx-{\int }_{\mathrm{\Omega }}{F}_v(x,u,v)\eta dx,\hfill \\ 〈J^{\mathrm{\prime }}(u,v),(\xi ,\eta )〉=-{\int }_{\mathrm{\Omega }}{G}_u(x,u,v)\xi dx-{\int }_{\mathrm{\Omega }}{G}_v(x,u,v)\eta dx.\hfill \end{array}$$

Hence, the weak solutions of problem (1.1) are exactly the solutions of the following equation:

$${\mathrm{\Phi }}^{\mathrm{\prime }}(u,v)+\lambda {\mathrm{\Psi }}^{\mathrm{\prime }}(u,v)+\mu J^{\mathrm{\prime }}(u,v)=0.$$

From (j3), for each $\lambda >0$, one has

$$\underset{\parallel (u,v)\parallel \to +\mathrm{\infty }}{lim}(\lambda \mathrm{\Phi }(u,v)+\mu \mathrm{\Psi }(u,v))=+\mathrm{\infty },$$

(2.8)

and so the first condition of Theorem 2.1 is satisfied. By Lemma 2.2, there exists $(u_0,{\upsilon }_0)\in X$ such that

$$\begin{array}{rl}\mathrm{\Phi }(u_0,v_0)& =\frac{{\hat{M}}_1({\parallel u_0\parallel }_p^p)}{p}+\frac{{\hat{M}}_2({\parallel v_0\parallel }_q^q)}{q}\\ \ge M_{-}({\parallel u_0\parallel }_p^p+{\parallel v_0\parallel }_q^q)=\frac{M_{-}}{C}[{(a{\alpha }_1)}^p+{(a{\alpha }_2)}^q]\\ >\frac{M_{-}}{C}\frac{b{M}^{+}}{M_{-}}=\frac{b{M}^{+}}{C}>0=\mathrm{\Phi }(0,0),\end{array}$$

(2.9)

and

$$|\mathrm{\Omega }|\underset{(x,s,t)\in \mathrm{\Omega }\times A(b{M}^{+}/M_{-})}{sup}F(x,s,t)\le \frac{b{M}^{+}}{C}\frac{{\int }_{\mathrm{\Omega }}F(x,u_0,{\upsilon }_0)dx}{\mathrm{\Phi }(u_0,{\upsilon }_0)}.$$

(2.10)

From (1.3), we have

$$\underset{x\in \overline{\mathrm{\Omega }}}{max}\left\{\right|u(x){|}^p\}\le C{\parallel u\parallel }_p^p,\underset{x\in \overline{\mathrm{\Omega }}}{max}\left\{\right|\upsilon (x){|}^q\}\le C{\parallel \upsilon \parallel }_q^q$$

for each $(u,\upsilon )\in X$. We obtain

$$\underset{x\in \overline{\mathrm{\Omega }}}{max}\{\frac{|u(x){|}^p}{p}+\frac{|v(x){|}^q}{q}\}\le C\{\frac{{\parallel u\parallel }_p^p}{p}+\frac{{\parallel v\parallel }_q^q}{q}\}$$

(2.11)

for each $(u,\upsilon )\in X$. Let $r=\frac{b{M}^{+}}{C}$ for each $(u,\upsilon )\in X$ such that

$$\mathrm{\Phi }(u,\upsilon )=\frac{{\hat{M}}_1({\parallel u\parallel }_p^p)}{p}+\frac{{\hat{M}}_2({\parallel v\parallel }_q^q)}{q}\le r.$$

From (2.11), we get

$$|u(x){|}^p+|\upsilon (x){|}^q\le C({\parallel u\parallel }_p^p+{\parallel \upsilon \parallel }_q^q)\le \frac{Cr}{M_{-}}=\frac{C}{M_{-}}\frac{b{M}^{+}}{C}=\frac{b{M}^{+}}{M_{-}}.$$

(2.12)

Then, from (2.10) and (2.12), we find

$$\begin{array}{rl}\underset{(u,\upsilon )\in {\mathrm{\Phi }}^{-1}(-\mathrm{\infty },r)}{sup}(-\mathrm{\Psi }(u,\upsilon ))& =\underset{\{(u,\upsilon )|\in \mathrm{\Phi }(u,\upsilon )\le r\}}{sup}{\int }_{\mathrm{\Omega }}F(x,u,\upsilon )dx\\ \le \underset{\{(u,\upsilon )||u(x){|}^p+|\upsilon (x){|}^q\le b{M}^{+}/M_{-}\}}{sup}{\int }_{\mathrm{\Omega }}F(x,u,\upsilon )dx\\ \le {\int }_{\mathrm{\Omega }}\underset{(s,t)\in A(b{M}^{+}/M_{-})}{sup}F(x,s,t)dx\\ \le |\mathrm{\Omega }|\underset{(x,s,t)\in \mathrm{\Omega }\times A(b{M}^{+}/M_{-})}{sup}F(x,s,t)\\ \le \frac{b{M}^{+}}{C}\frac{{\int }_{\mathrm{\Omega }}F(x,u_0,{\upsilon }_0)dx}{\mathrm{\Phi }(u_0,{\upsilon }_0)}\\ =r\frac{-\mathrm{\Psi }(u_0,{\upsilon }_0)}{\mathrm{\Phi }(u_0,{\upsilon }_0)}.\end{array}$$

Hence, we have

$$\underset{\{(u,v)|\mathrm{\Phi }(u,v\le r\}}{sup}(-\mathrm{\Psi }(u,v))<r\frac{-\mathrm{\Psi }(u_0,{\upsilon }_0)}{\mathrm{\Phi }(u_0,{\upsilon }_0)}.$$

(2.13)

Fix h such that

$$\underset{\{(u,v)|\mathrm{\Phi }(u,v\le r\}}{sup}(-\mathrm{\Psi }(u,v))<h<r\frac{-\mathrm{\Psi }(u_0,{\upsilon }_0)}{\mathrm{\Phi }(u_0,{\upsilon }_0)},$$

by (2.9), (2.13) and Proposition 2.1, with $(u_1,v_1)=(0,0)$ and $(u^{\ast },v^{\ast })=(u_0,v_0)$, we obtain

$$\underset{\lambda \ge 0}{sup}\underset{x\in X}{inf}(\mathrm{\Phi }(x)+\lambda (h+\mathrm{\Psi }(x)))<\underset{x\in X}{inf}\underset{\lambda \ge 0}{sup}(\mathrm{\Phi }(x)+\lambda (h+\mathrm{\Psi }(x))),$$

(2.14)

and so assumption (2.1) of Theorem 2.1 is satisfied.

Now, with $I=[0,\mathrm{\infty })$, from (2.8) and (2.14), all the assumptions of Theorem 2.1 hold. Hence, our conclusion follows from Theorem 2.1. □