1 Introduction and main results

In this paper, we consider the existence of weak solutions for the following gradient elliptic systems:

$$ \textstyle\begin{cases} -\triangle_{p}u=\lambda_{1} a(x)\vert u\vert ^{p-2}u+\lambda_{1}\frac{b(x)}{\beta+1} \vert u\vert ^{\alpha} \vert v\vert ^{\beta}v+F_{u}(x,u,v)-h_{1}(x) & \mbox{in } \Omega ,\\ -\triangle_{p}v=\lambda_{1} c(x)\vert v\vert ^{p-2}v+\lambda_{1}\frac{b(x)}{\alpha+1} \vert u\vert ^{\alpha} \vert v\vert ^{\beta}u+F_{v}(x,u,v)-h_{2}(x) & \mbox{in } \Omega,\\ u=v=0 &\mbox{on } \partial\Omega, \end{cases} $$
(1)

where \(\Omega\subset R^{N}\) (\(N\geq3\)) is a bounded smooth domain, \(\triangle_{p}u=\operatorname{div}(\vert \nabla u\vert ^{p-2}\nabla u)\) denotes the p-Laplacian, \(2\leq p< N\) and \(\alpha\geq0\), \(\beta\geq0\) satisfy

$$\alpha+\beta+2=p. $$

\(F\in C^{1}(\overline{\Omega}\times R^{2},R)\) and \(F_{s}(x,s,t)\) designates the partial derivative of F with respect to s and \(h_{1},h_{2}\in L^{q}(\Omega)\) (\(q=p/(p-1)\)). The coefficient functions \(a,b,c\in C(\Omega)\cap L^{\infty}(\Omega)\) satisfy one of the following conditions:

  1. (A1)

    \(a^{+}\neq0\), where \(a^{+}(x):=\max\{a(x),0\}\);

  2. (A2)

    \(c^{+}\neq0\);

  3. (A3)

    \(a=c=0 \mbox{ and }b^{+}\neq0\).

Let W be the product space \(W^{1,p}_{0}(\Omega)\times W^{1,p}_{0}(\Omega)\) equipped with the norm \(\Vert (u,v)\Vert =(\Vert u\Vert ^{p}+\Vert v\Vert ^{p})^{1/p}\) for all \((u,v)\in W\), where \(\Vert u\Vert = (\int_{\Omega} \vert \nabla u\vert ^{p}\,dx )^{1/p}\) for any \(u\in W^{1,p}_{0}(\Omega)\). The embedding \(W^{1,p}_{0}(\Omega)\hookrightarrow L^{p}(\Omega)\) is continuous and there exists a positive constant C such that

$$ \Vert u\Vert _{L^{p}}\leq C\Vert u\Vert \quad \mbox{for all }u\in W^{1,p}_{0}(\Omega), $$
(2)

where \(\Vert \cdot \Vert _{L^{p}}\) denotes the norm of \(L^{p}(\Omega)\).

Consider the following nonlinear eigenvalue problem with weights:

$$ \textstyle\begin{cases} -\triangle_{p}u=\lambda a(x)\vert u\vert ^{p-2}u+\lambda\frac{b(x)}{\beta+1} \vert u\vert ^{\alpha} \vert v\vert ^{\beta}v & \mbox{in } \Omega,\\ -\triangle_{p}v=\lambda c(x)\vert v\vert ^{p-2}v+\lambda\frac{b(x)}{\alpha+1} \vert u\vert ^{\alpha} \vert v\vert ^{\beta}u & \mbox{in } \Omega,\\ u=v=0 & \mbox{on } \partial\Omega. \end{cases} $$
(3)

If one of the conditions (A1)-(A3) holds, the first eigenvalue \(\lambda _{1}\) of (3) is simple, isolated and positive, and has a unique associated eigenfunction \((\mu_{1},\nu_{1})\) with \(\Vert (\mu_{1},\nu_{1})\Vert =1\) and \(\mu_{1}>0\), \(\nu_{1}>0\) in Ω (the proof is found in [1, 2]).

The Landesman-Lazer-type conditions were introduced by Landesman and Lazer in [3], where they considered the existence of weak solutions for the resonant elliptic problems, and then were widely used and extended (see [1–10] and their references). For nonlinear elliptic systems, let \(F_{s}(x,s,t)=g_{1}(s)\), \(F_{t}(x,s,t)=g_{2}(t)\) and by using the some Landesman-Lazer-type conditions, Zographopoulos in [1] proved the existence of weak solutions for problem (1) at resonance with the first eigenvalue \(\lambda_{1}\), and by using the Landesman-Lazer-type conditions due to Tang and the G-linking theorem, Ou and Tang in [2] proved the existence of weak solutions for problem (1) at resonance with the higher eigenvalues of problem (3). When \(p=2\), Silva in [10] introduced the new Landesman-Lazer-type conditions and proved the existence of weak solutions for problem (1) by using variational methods, Morse theory and critical groups.

Motivated by [10], we consider the existence of weak solutions for problem (1) under the certain Landesman-Lazer-type conditions. We now give some auxiliary conditions.

  1. (F1)

    There is \(h\in C(\Omega, R^{+})\) such that

    $$\bigl\vert F_{s}(x,s,t)\bigr\vert \leq h(x) \quad \mbox{and}\quad \bigl\vert F_{t}(x,s,t)\bigr\vert \leq h(x), \quad \forall (x,s,t)\in\Omega \times R^{2}. $$
  2. (F2)

    There exist functions \(f^{++}, f^{--}\in C(\Omega, R)\) such that

    $$\begin{aligned} f^{++}(x)=\mathop{\lim_{s\to+\infty}}_{t\to+\infty}F_{s}(x,s,t),\qquad f^{--}(x)=\mathop{\lim_{s\to-\infty}}_{t\to-\infty}F_{s}(x,s,t). \end{aligned}$$
  3. (F3)

    There exist functions \(g^{++},g^{--}\in C(\Omega, R)\) such that

    $$\begin{aligned} g^{++}(x)=\mathop{\lim_{s\to+\infty}}_{t\to+\infty} F_{t}(x,s,t), \qquad g^{--}(x)=\mathop{\lim_{s\to+\infty}}_{t\to+\infty} F_{t}(x,s,t), \end{aligned}$$

    where the above limits of conditions (F2) and (F3) are taken uniformly for all \(x\in\Omega\). The Landesman-Lazer-type conditions for problem (1) will be assumed either

    $$(LL)^{+}_{1} \quad \int_{\Omega}f^{--} \mu_{1}+g^{--}\nu_{1}\,dx< \int_{\Omega}h_{1}\mu_{1}+h_{2}\nu_{1}\,dx< \int _{\Omega}f^{++}\mu_{1}+g^{++} \nu_{1}\,dx $$

    or

    $$(LL)^{-}_{1}\quad \int_{\Omega}f^{--} \mu_{1}+g^{--}\nu_{1}\,dx> \int_{\Omega}h_{1}\mu_{1}+h_{2}\nu_{1}\,dx>\int _{\Omega}f^{++}\mu_{1}+g^{++} \nu_{1}\,dx. $$

We are ready to introduce the main results of this paper.

Theorem 1

Assume that \(h_{1},h_{2}\in L^{q}(\Omega)\) (\(q=p/(p-1)\)) and one of the conditions (A1)-(A3) holds. If F satisfies (F1), (F2), (F3) and \((LL)^{+}_{1}\), then problem (1) has at least one solution.

Theorem 2

Assume that \(h_{1},h_{2}\in L^{q}(\Omega)\) (\(q=p/(p-1)\)) and one of the conditions (A1)-(A3) holds. If F satisfies (F1), (F2), (F3) and \((LL)^{-}_{1}\), then problem (1) has at least one solution.

2 Proofs of theorems

Let \(J: W\to R\) be the functional defined by

$$ J(u,v)=\phi(u,v)-\lambda_{1}\psi(u,v)-\int _{\Omega}F(x,u,v)\,dx+\int_{\Omega}h_{1}(x)u\,dx+\int_{\Omega}h_{2}(x)v\,dx, $$
(4)

where

$$\begin{aligned}& \phi(u,v)=\frac{1}{p}\int_{\Omega} \vert \nabla u\vert ^{p}\,dx +\frac{1}{p}\int_{\Omega} \vert \nabla v\vert ^{p}\,dx, \quad \mbox{and} \\& \psi(u,v)=\frac{1}{p}\int_{\Omega}a(x)\vert u\vert ^{p}\,dx +\frac{1}{p}\int_{\Omega}c(x)\vert v \vert ^{p}\,dx+ \frac{1}{(\alpha+1)(\beta+1)}\int_{\Omega}b(x) \vert u\vert ^{\alpha} \vert v\vert ^{\beta}uv\,dx. \end{aligned}$$

If one of the conditions (A1)-(A3) holds, by (F1) and \(h_{1},h_{2}\in L^{q}(\Omega)\), it is not difficult to verify that \(J\in C^{1}(W,R)\), and it is well known that a critical point of the functional J in W corresponds to a weak solution of problem (1). We will prove Theorem 1 by the saddle point theorem due to Rabinowitz (see [11]) and Theorem 2 by Ekeland’s variational principle (see [12]).

Proof of Theorem 1

We divide the proof into two steps.

(i) We claim that the functional J satisfies the \((PS)\) condition. Let \((u_{n},v_{n})\in W\) be a \((PS)\) sequence for the functional J, that is,

$$ J(u_{n},v_{n})\to c\in R\quad \mbox{and}\quad J'(u_{n},v_{n})\to0 \quad \mbox{as } n\to\infty. $$
(5)

We first verify that \((u_{n},v_{n})\) is bounded in W, and then prove that \((u_{n},v_{n})\) has a convergent subsequence. Suppose, by contradiction, that \(K_{n}:= \Vert (u_{n},v_{n})\Vert =(\Vert u_{n}\Vert ^{p}+\Vert v_{n}\Vert ^{p})^{1/p}\to\infty\) as \(n\to\infty\). Let \(\tilde{u}_{n}=u_{n}\setminus K_{n}\), \(\tilde{v}_{n}=v_{n}\setminus K_{n}\), then \((\tilde{u}_{n},\tilde{v}_{n})\) is bounded in W, that is,

$$\Vert \tilde{u}_{n}\Vert ^{p}+\Vert \tilde{v}_{n}\Vert ^{p}=1\quad \mbox{for all }n. $$

Hence there is a subsequence of \((\tilde{u}_{n},\tilde{v}_{n})\), still denoted by \((\tilde{u}_{n},\tilde{v}_{n})\), and \((\tilde{u},\tilde{v})\in W\) such that \((\tilde{u}_{n},\tilde{v}_{n})\rightharpoonup(\tilde{u},\tilde{v})\) weakly in W, \((\tilde{u}_{n},\tilde{v}_{n})\to(\tilde{u},\tilde{v})\) strongly in \(L^{p}(\Omega)\times L^{p}(\Omega)\) and \((\tilde{u}_{n}(x),\tilde{v}_{n}(x))\to(\tilde{u}(x),\tilde{v}(x))\) for a.e. \(x\in\Omega\). From (F1), (2) and Hölder’s inequality, we obtain

$$\begin{aligned} \biggl\vert \int_{\Omega}F(x,u,v)\,dx\biggr\vert \leq&\int_{\Omega}\bigl\vert F(x,u,v)\bigr\vert \,dx \\ =&\int_{\Omega}\bigl\vert F(x,u,v)-F(x,0,0)+F(x,0,0)\bigr\vert \,dx \\ \leq&\int_{\Omega}\biggl\vert \int_{0}^{1} \bigl(F_{s}(x,\tau u,\tau v)u+F_{t}(x,\tau u,\tau v)v\bigr) \,d\tau\biggr\vert \,dx+\int_{\Omega}\bigl\vert F(x,0,0)\bigr\vert \,dx \\ \leq&\int_{\Omega}h(x) \bigl(\vert u\vert +\vert v \vert \bigr)\,dx+C_{0} \\ \leq& C\Vert h\Vert _{L^{q}}\bigl(\Vert u\Vert +\Vert v\Vert \bigr)+C_{0} \end{aligned}$$
(6)

for all \((u,v)\in W\), where \(C_{0}=\int_{\Omega} \vert F(x,0,0)\vert \,dx\), hence we get

$$ \frac{1}{\Vert u_{n}\Vert ^{p}+\Vert v_{n}\Vert ^{p}}\int_{\Omega}F(x,u_{n},v_{n}) \,dx\to0 \quad \mbox{as }n\to\infty, $$
(7)

and from \(h_{1},h_{2}\in L^{q}(\Omega)\) (\(q=p/(p-1)\)) and Hölder’s inequality, it follows that

$$ \frac{1}{\Vert u_{n}\Vert ^{p}+\Vert v_{n}\Vert ^{p}}\int_{\Omega}(h_{1}u_{n}+h_{2}v_{n}) \,dx\to0 \quad \mbox{as }n\to\infty. $$
(8)

From \((\tilde{u}_{n},\tilde{v}_{n})\to(\tilde{u},\tilde{v})\) strongly in \(L^{p}(\Omega)\times L^{p}(\Omega)\), we have \(\vert \tilde{u}_{n}\vert ^{p}\to \vert \tilde{u}\vert ^{p}\) and \(\vert \tilde{v}_{n}\vert ^{p}\to \vert \tilde{v}\vert ^{p}\) strongly in \(L^{1}(\Omega)\times L^{1}(\Omega)\). Hence, it follows that

$$ \biggl\vert \int_{\Omega}a(x)\vert \tilde{u}_{n}\vert ^{p}\,dx-\int_{\Omega}a(x)\vert \tilde {u}\vert ^{p}\,dx\biggr\vert \leq \Vert a \Vert _{L^{\infty}}\int_{\Omega}\bigl\vert \vert \tilde {u}_{n}\vert ^{p}-\vert \tilde{u}\vert ^{p} \bigr\vert \,dx\to0 $$
(9)

as \(n\to\infty\).

From \((\tilde{u}_{n}(x),\tilde{v}_{n}(x))\to(\tilde{u}(x),\tilde{v}(x))\) for a.e. \(x\in\Omega\) and

$$\begin{aligned}& \int_{\Omega}\bigl\vert \vert \tilde{u}_{n} \vert ^{\alpha}\tilde{u}_{n}\bigr\vert ^{\frac{p}{\alpha+1}}\,dx = \Vert \tilde{u}_{n}\Vert _{L^{p}}^{p}\to \Vert \tilde{u}\Vert _{L^{p}}^{p} =\int_{\Omega}\bigl\vert \vert \tilde{u}\vert ^{\alpha}\tilde{u}\bigr\vert ^{\frac{p}{\alpha+1}}\,dx, \\& \int_{\Omega}\bigl\vert \vert \tilde{v}_{n} \vert ^{\beta}\tilde{v}_{n}\bigr\vert ^{\frac{p}{\beta+1}}\,dx = \Vert \tilde{v}_{n}\Vert _{L^{p}}^{p}\to \Vert \tilde{v}\Vert _{L^{p}}^{p} =\int_{\Omega}\bigl\vert \vert \tilde{v}\vert ^{\beta}\tilde{v}\bigr\vert ^{\frac{p}{\beta+1}}\,dx \end{aligned}$$

as \(n\to\infty\), it follows that \(\vert \tilde{u}_{n}\vert ^{\alpha}\tilde{u}_{n}\to \vert \tilde{u}\vert ^{\alpha}\tilde{u}\) strongly in \(L^{\frac{p}{\alpha+1}}(\Omega )\) and \(\vert \tilde{v}_{n}\vert ^{\beta}\tilde{v}_{n}\to \vert \tilde{v}\vert ^{\beta}\tilde {v}\) strongly in \(L^{\frac{p}{\beta+1}}(\Omega)\). Hence from Hölder’s inequality we obtain

$$\begin{aligned} &\biggl\vert \int_{\Omega}b(x) \bigl(\vert \tilde{u}_{n}\vert ^{\alpha} \vert \tilde{v}_{n} \vert ^{\beta}\tilde{u}_{n}\tilde{v}_{n}- \vert \tilde{u}\vert ^{\alpha} \vert \tilde{v}\vert ^{\beta}\tilde{u} \tilde{v}\bigr)\,dx\biggr\vert \\ &\quad \leq \Vert b\Vert _{L^{\infty}}\int_{\Omega}\bigl\vert \vert \tilde{u}_{n}\vert ^{\alpha} \vert \tilde{v}_{n}\vert ^{\beta}\tilde{u}_{n} \tilde{v}_{n}- \vert \tilde{u}_{n}\vert ^{\alpha} \vert \tilde{v}\vert ^{\beta}\tilde{u}_{n}\tilde {v}\bigr\vert \,dx \\ &\qquad {}+\Vert b\Vert _{L^{\infty}}\int_{\Omega}\bigl\vert \vert \tilde{u}_{n}\vert ^{\alpha} \vert \tilde{v}\vert ^{\beta}\tilde{u}_{n}\tilde{v}- \vert \tilde{u}\vert ^{\alpha} \vert \tilde{v}\vert ^{\beta}\tilde{u}\tilde{v}\bigr\vert \,dx \\ &\quad \leq \Vert b\Vert _{L^{\infty}}\int_{\Omega} \vert \tilde{u}_{n}\vert ^{\alpha+1}\cdot\bigl\vert \vert \tilde{v}_{n}\vert ^{\beta}\tilde{v}_{n}-\vert \tilde{v}\vert ^{\beta}\tilde {v}\bigr\vert \,dx \\ &\qquad {}+\Vert b\Vert _{L^{\infty}}\int_{\Omega}\bigl\vert \vert \tilde{u}_{n}\vert ^{\alpha}\tilde{u}_{n}- \vert \tilde{u}\vert ^{\alpha}\tilde{u}\bigr\vert \cdot \vert \tilde{v}\vert ^{\beta+1}\,dx \\ &\quad \leq \Vert b\Vert _{L^{\infty}} \Vert \tilde{u}_{n}\Vert _{L^{p}}^{\alpha+1}\bigl\Vert \vert \tilde{v}_{n} \vert ^{\beta}\tilde{v}_{n}-\vert \tilde{v}\vert ^{\beta}\tilde{v}\bigr\Vert _{L^{\frac{p}{\beta+1}}} \\ &\qquad {}+\Vert b\Vert _{L^{\infty}} \Vert \tilde{v}_{n}\Vert _{L^{p}}^{\beta+1} \bigl\Vert \vert \tilde{u}_{n} \vert ^{\alpha}\tilde{u}_{n}-\vert \tilde{u}\vert ^{\alpha}\tilde{u}\bigr\Vert _{L^{\frac{p}{\alpha+1}}} \\ &\quad \to0 \quad \mbox{as }n\to\infty. \end{aligned}$$
(10)

From (5) it follows that

$$\limsup_{n\to\infty}\frac{J(u_{n},v_{n})}{K^{p}_{n}}\leq0. $$

Combining the above inequality with (7), (8), (9) (10) and \(\alpha+\beta+2=p\), we have

$$\begin{aligned} &\limsup_{n\to\infty} \biggl(\int_{\Omega} \vert \nabla\tilde{u}_{n}\vert ^{p}\,dx +\int _{\Omega} \vert \nabla\tilde{v}_{n}\vert ^{p}\,dx \biggr) \\ &\quad \leq\lambda_{1} \biggl(\int_{\Omega}a(x)\vert \tilde{u}\vert ^{p}\,dx +\int_{\Omega}c(x)\vert \tilde{v}\vert ^{p}\,dx+ \frac{p}{(\alpha+1)(\beta+1)}\int_{\Omega}b(x) \vert \tilde{u}\vert ^{\alpha} \vert \tilde{v}\vert ^{\beta} \tilde{u}\tilde{v}\,dx \biggr). \end{aligned}$$

Hence, using the weak lower semicontinuity of the norm and the Poincaré inequality, we obtain

$$\begin{aligned} &\lambda_{1} \biggl(\int_{\Omega}a(x)\vert \tilde{u}\vert ^{p}\,dx +\int_{\Omega}c(x)\vert \tilde{v}\vert ^{p}\,dx+ \frac{p}{(\alpha+1)(\beta+1)}\int_{\Omega}b(x) \vert \tilde{u}\vert ^{\alpha} \vert \tilde{v}\vert ^{\beta} \tilde{u}\tilde{v}\,dx \biggr) \\ &\quad \leq\int_{\Omega} \vert \nabla\tilde{u}\vert ^{p} \,dx +\int_{\Omega} \vert \nabla\tilde{v}\vert ^{p} \,dx \\ &\quad \leq\liminf_{n\to\infty} \biggl(\int_{\Omega} \vert \nabla\tilde{u}_{n}\vert ^{p}\,dx +\int _{\Omega} \vert \nabla\tilde{v}_{n}\vert ^{p}\,dx \biggr) \\ &\quad \leq\limsup_{n\to\infty} \biggl(\int_{\Omega} \vert \nabla\tilde{u}_{n}\vert ^{p}\,dx +\int _{\Omega} \vert \nabla\tilde{v}_{n}\vert ^{p}\,dx \biggr) \\ &\quad \leq\lambda_{1} \biggl(\int_{\Omega}a(x)\vert \tilde{u}\vert ^{p}\,dx +\int_{\Omega}c(x)\vert \tilde{v}\vert ^{p}\,dx+ \frac{p}{(\alpha+1)(\beta+1)}\int_{\Omega}b(x) \vert \tilde{u}\vert ^{\alpha} \vert \tilde{v}\vert ^{\beta} \tilde{u}\tilde{v}\,dx \biggr), \end{aligned}$$

which implies that the following equality holds:

$$\begin{aligned} &\int_{\Omega} \vert \nabla\tilde{u}\vert ^{p}\,dx +\int_{\Omega} \vert \nabla\tilde{v}\vert ^{p}\,dx \\ &\quad =\lambda_{1} \biggl(\int_{\Omega}a(x)\vert \tilde{u}\vert ^{p}\,dx +\int_{\Omega}c(x)\vert \tilde{v}\vert ^{p}\,dx+ \frac{p}{(\alpha+1)(\beta+1)}\int_{\Omega}b(x) \vert \tilde{u}\vert ^{\alpha} \vert \tilde{v}\vert ^{\beta} \tilde{u}\tilde{v}\,dx \biggr). \end{aligned}$$

By the uniform convexity of W, we have that \((\tilde{u}_{n},\tilde{v}_{n})\) converges strongly to \((\tilde{u},\tilde{v})\) in W, and from the definition of \((\mu_{1},\nu_{1})\), it follows that \((\tilde{u},\tilde{v})=\pm(\mu_{1},\nu_{1})\).

In the following, we assume that \((\tilde{u},\tilde{v})=(\mu_{1},\nu_{1})\), and the case where \((\tilde{u},\tilde{v})=-(\mu_{1},\nu_{1})\) may be treated similarly. Noting that \(\alpha+\beta+2=p\), it follows that

$$\begin{aligned} &\frac{p}{K_{n}(\alpha+1)(\beta+1)}\int_{\Omega}b(x)\vert u_{n} \vert ^{\alpha} \vert v_{n}\vert ^{\beta}u_{n}v_{n} \,dx \\ &\quad =\frac{1}{\beta+1}\int_{\Omega}b(x)\vert u_{n} \vert ^{\alpha} \vert v_{n}\vert ^{\beta} \tilde{u}_{n}v_{n} \,dx+ \frac{1}{\alpha+1}\int _{\Omega}b(x)\vert u_{n}\vert ^{\alpha} \vert v_{n}\vert ^{\beta}u_{n} \tilde{v}_{n} \,dx. \end{aligned}$$

Hence from (4) and the above equality, we have

$$\begin{aligned} &\frac{pJ(u_{n},v_{n})}{K_{n}}-\bigl\langle J'(u_{n},v_{n}), (\tilde{u}_{n},\tilde{v}_{n})\bigr\rangle \\ &\quad =\int_{\Omega}\bigl(F_{s}(x,u_{n},v_{n}) \tilde{u}_{n}+F_{t}(x,u_{n},v_{n}) \tilde{v}_{n}\bigr)\,dx- \frac{p}{K_{n}}\int_{\Omega}F(x,u_{n},v_{n})\,dx \\ &\qquad {}+(p-1)\int_{\Omega}(h_{1}\tilde{u}_{n}+h_{2} \tilde{v}_{n})\,dx. \end{aligned}$$
(11)

From \(h_{1}, h_{2}\in L^{q}(\Omega)\), we observe

$$ \int_{\Omega}h_{1}\tilde{u}_{n}+h_{2} \tilde{v}_{n}\,dx\to\int_{\Omega}h_{1} \mu_{1}+h_{2}\nu_{1} \,dx \quad \mbox{as } n\to\infty. $$
(12)

From (F2) and (F3), we have

$$\begin{aligned} \int_{\Omega}F_{s}(x,u_{n},v_{n}) \tilde{u}_{n}+F_{t}(x,u_{n},v_{n}) \tilde{v}_{n}\,dx \to\int_{\Omega}\bigl(f^{++} \mu_{1}+g^{++}\nu_{1}\bigr)\,dx \quad \mbox{as } n\to \infty. \end{aligned}$$
(13)

Finally, from the Lebesgue dominated convergence theorem, (F2) and (F3), we have

$$\begin{aligned} &\frac{1}{K_{n}}\int_{\Omega}F(x,u_{n},v_{n})\,dx \\ &\quad =\frac{1}{K_{n}}\int_{\Omega}\int_{0}^{1} \bigl(F_{s}(x,\tau u_{n},\tau v_{n})u_{n}+F_{t}(x, \tau u_{n},\tau v_{n})v_{n}\bigr)\,d\tau\,dx+ \frac{C_{0}}{K_{n}} \\ &\quad =\int_{\Omega}\int_{0}^{1} \bigl(F_{s}(x,\tau u_{n},\tau v_{n}) \tilde{u}_{n}+F_{t}(x,\tau u_{n},\tau v_{n})\tilde{v}_{n}\bigr)\,d\tau\,dx+\frac {C_{0}}{K_{n}} \\ &\quad \to\int_{\Omega}\bigl(f^{++}\mu_{1}+g^{++} \nu_{1}\bigr)\,dx \quad \mbox{as } n\to\infty. \end{aligned}$$
(14)

Therefore, taking the limit in (11) and from (5), (12), (13) and (14), we get

$$\begin{aligned} \int_{\Omega}(h_{1}\mu_{1}+h_{2} \nu_{1})\,dx=\int_{\Omega}\bigl(f^{++} \mu_{1}+g^{++}\nu_{1}\bigr)\,dx, \end{aligned}$$

which is a contradiction with the condition \((LL)^{+}_{1}\). Hence, \((u_{n},v_{n})\) is bounded in W, and there is a subsequence of \((u_{n},v_{n})\) without any loss of generality still denoted by \((u_{n},v_{n})\), and \((u,v)\in W\) such that \((u_{n},v_{n})\rightharpoonup(u,v)\) weakly in W, \((u_{n},v_{n})\to(u,v)\) strongly in \(L^{p}(\Omega)\times L^{p}(\Omega)\). Consequently, from (5), one has

$$ \lim_{n\to\infty}\bigl\langle J'(u_{n},v_{n}),(u_{n}-u,0) \bigr\rangle =0. $$
(15)

From (F1) and Hölder’s inequality, it follows that

$$\biggl\vert \int_{\Omega}F_{s}(x,u_{n},v_{n}) ({u}_{n}-u)\,dx\biggr\vert \leq \Vert h\Vert _{L^{q}} \Vert u_{n}-u\Vert _{L^{p}}\to0 $$

as \(n\to\infty\). Similarly, we obtain

$$\biggl\vert \int_{\Omega}h_{1}(x) ({u}_{n}-u)\,dx\biggr\vert \leq \Vert h_{1}\Vert _{L^{q}}\Vert u_{n}-u\Vert _{L^{p}}\to0 $$

and

$$\begin{aligned} \biggl\vert \int_{\Omega}a(x) \vert {u}_{n} \vert ^{p-2}{u}_{n}({u}_{n}-{u})\,dx\biggr\vert &\leq \Vert a\Vert _{L^{\infty}} \Vert {u}_{n}\Vert ^{p-1}_{L^{p}}\Vert {u}_{n}-{u}\Vert _{L^{p}} \\ &\leq C^{p-1}\Vert a\Vert _{L^{\infty}} \Vert {u}_{n} \Vert ^{p-1}\Vert {u}_{n}-{u}\Vert _{L^{p}} \\ &\to 0 \end{aligned}$$

as \(n\to\infty\). Combining the above three inequalities and (15), we get

$$\int_{\Omega}\bigl(\vert \nabla{u}_{n}\vert ^{p-2}\nabla{u}_{n}, \nabla({u}_{n}-{u})\bigr)\,dx \to0 $$

as \(n\to\infty\). Similarly, we also obtain

$$\lim_{n\to\infty}\int_{\Omega}\bigl(\vert \nabla u \vert ^{p-2}\nabla u,\nabla (u_{n}-u)\bigr)\,dx=0, $$

hence

$$\lim_{n\to\infty}\int_{\Omega}\bigl(\bigl(\vert \nabla u_{n}\vert ^{p-2}\nabla u_{n}-\vert \nabla u\vert ^{p-2}\nabla u\bigr),\nabla(u_{n}-u)\bigr) \,dx=0. $$

From Clarkson’s inequality, that is, there is \(C_{p}>0\) such that for all \(\mu,\nu\in R^{N}\) and \(p\geq2\),

$$\vert \mu-\nu \vert ^{p} \leq C_{p}\bigl(\vert \mu \vert ^{p-2}\mu-\vert \nu \vert ^{p-2}\nu\bigr) (\mu-\nu), $$

it follows that

$$\lim_{n\to\infty}\int_{\Omega} \vert \nabla u_{n}-\nabla u\vert ^{p}\,dx=0, $$

this is, \(u_{n}\to u\) in \(W^{1,p}_{0}(\Omega)\). Similarly, we have \(v_{n}\to v\) in \(W^{1,p}_{0}(\Omega)\), hence \((u_{n},v_{n})\to(u,v)\) strongly in W.

(ii) We claim that the functional J satisfies the geometries of the saddle point theorem with respect to \((E_{1},E_{2})\), where \(E_{1}=\operatorname {span}\{(\mu_{1},\nu_{1})\}\), \(E_{2}=\{(\phi,\psi)\in W:\int_{\Omega}(\mu _{1}^{p-1}\phi+\nu_{1}^{p-1}\psi)\,dx=0\}\) and \(W=E_{1}\oplus E_{2}\).

By the definition of \((\mu_{1},\nu_{1})\), for all \(t\in R\), we have

$$\begin{aligned} &\lambda_{1} \biggl(\int_{\Omega}a(x) \vert t\mu_{1}\vert ^{p}\,dx +\int_{\Omega}c(x)\vert t\nu_{1}\vert ^{p}\,dx \\ &\qquad {}+ \frac{p}{(\alpha+1)(\beta+1)}\int _{\Omega}b(x)\vert t\mu_{1}\vert ^{\alpha} \vert t\nu_{1}\vert ^{\beta} t\mu_{1}t \nu_{1}\,dx \biggr) \\ &\quad =\int_{\Omega}\bigl\vert \nabla(t\mu_{1})\bigr\vert ^{p}\,dx +\int_{\Omega}\bigl\vert \nabla(t \nu_{1})\bigr\vert ^{p}\,dx. \end{aligned}$$
(16)

Moreover, we have

$$\begin{aligned} &\int_{\Omega}F(x,t\mu_{1},t \nu_{1})\,dx \\ &\quad =\int_{\Omega}\bigl(F(x,t\mu_{1},t \nu_{1})-F(x,0,0)\bigr)\,dx+\int_{\Omega}F(x,0,0)\,dx \\ &\quad =\int_{\Omega}\int_{0}^{1} \bigl(F_{s}(x,\tau t\mu_{1},\tau t\nu_{1})t \mu_{1}+F_{t}(x,\tau t\mu_{1},\tau t \nu_{1})t\nu_{1}\bigr)\,d\tau \,dx+\int_{\Omega}F(x,0,0)\,dx \\ &\quad =t\int_{\Omega}\int_{0}^{1} \bigl(F_{s}(x,\tau t\mu_{1},\tau t\nu_{1}) \mu_{1}+F_{t}(x,\tau t\mu_{1},\tau t \nu_{1})\nu_{1}\bigr)\,d\tau\,dx+\int_{\Omega}F(x,0,0)\,dx. \end{aligned}$$
(17)

From the Lebesgue dominated convergence theorem, (F1), (F2) and (F3), we obtain

$$\begin{aligned} &\lim_{t\to+\infty}\int_{\Omega}\int _{0}^{1}\bigl(F_{s}(x,\tau t \mu_{1},\tau t\nu_{1})\mu_{1}+F_{t}(x, \tau t\mu_{1},\tau t\nu_{1})\nu_{1}\bigr)\,d\tau \,dx \\ &\quad =\int_{\Omega}\bigl(f^{++}\mu_{1}+g^{++} \nu_{1}\bigr)\,dx. \end{aligned}$$
(18)

Hence, from (4), \((LL)^{+}_{1}\), (16), (17) and (18), it follows that

$$\begin{aligned} J(t\mu_{1},t\nu_{1})&= t\int_{\Omega}(h_{1}\mu_{1}+h_{2}\nu_{1})\,dx-\int _{\Omega}F(x,t\mu_{1},t\nu_{1})\,dx \\ &\to -\infty \quad \mbox{as } t\to\infty. \end{aligned}$$

Similarly, if t tends to −∞, the same result is obtained with \(f^{++}\) and \(g^{++}\) exchanged with \(f^{--}\) and \(g^{--}\) respectively. Hence, in both cases we have

$$ \lim_{\vert t\vert \to\infty}J(t\mu_{1},t \nu_{1})=-\infty. $$
(19)

On the other hand, from the definition of \(\lambda_{1}\), there is \(\bar {\lambda}>\lambda_{1}\) such that

$$\begin{aligned} &\int_{\Omega} \vert \nabla u\vert ^{p}\,dx +\int _{\Omega} \vert \nabla v\vert ^{p}\,dx \\ &\quad \geq\bar{\lambda} \biggl(\int_{\Omega}a(x)\vert u\vert ^{p}\,dx +\int_{\Omega}c(x)\vert v\vert ^{p}\,dx+ \frac{p}{(\alpha+1)(\beta+1)}\int_{\Omega}b(x)\vert u \vert ^{\alpha} \vert v\vert ^{\beta} uv\,dx \biggr) \end{aligned}$$

for all \((u,v)\in E_{2}\). From (2), (4), (6), the above inequality and Hölder’s inequality, we obtain

$$\begin{aligned} J(u,v) \geq&\frac{\bar{\lambda}-\lambda_{1}}{p\bar{\lambda}} \bigl(\Vert u\Vert ^{p}+\Vert v\Vert ^{p}\bigr)-C\Vert h\Vert _{L^{q}}\bigl(\Vert u\Vert +\Vert v\Vert \bigr) \\ &{}-\bigl(\Vert h_{1}\Vert _{L^{q}}\Vert u\Vert _{L^{p}}+\Vert h_{2}\Vert _{L^{q}}\Vert v\Vert _{L^{p}}\bigr)-C_{0} \\ \geq&\frac{\bar{\lambda}-\lambda_{1}}{p\bar{\lambda}}\bigl(\Vert u\Vert ^{p}+\Vert v\Vert ^{p}\bigr) -C_{1}\bigl(\Vert u\Vert +\Vert v\Vert \bigr)-C_{0} \end{aligned}$$
(20)

for all \((u,v)\in E_{2}\), where \(C_{1}=C(\Vert h\Vert _{L^{q}}+\min\{\Vert h_{1}\Vert _{L^{q}}, \Vert h_{2}\Vert _{L^{q}}\})\).

Thus, from (19) and (20), there is \(\delta\in R\) and \(R_{0}>0\) such that if \(\vert t\vert =R_{0}\) we obtain

$$J(t\mu_{1},t\nu_{1})< \delta< \min_{(u,v)\in E_{2}}J(u,v). $$

From the saddle point theorem, Theorem 1 is proved. □

Proof of Theorem 2

(i) Similar to (i) of the proof of Theorem 1, we can prove that from \((LL)^{-}_{1}\), the functional J satisfies the \((PS)\) condition.

(ii) Now we will prove that the functional J is coercive, that is,

$$J(u,v)\to+\infty \quad \mbox{as } \bigl\Vert (u,v)\bigr\Vert \to\infty. $$

If the claim does not hold, there is a constant c and a sequence \((u_{n},v_{n})\) with \(\Vert (u_{n},v_{n})\Vert \to\infty\) as \(n\to\infty\) such that \(J(u_{n},v_{n})\leq c\). Let \(K_{n}:=(\Vert u_{n}\Vert ^{p}+\Vert v_{n}\Vert ^{p})^{1/p}\), hence we have \(K_{n}\to\infty\) as \(n\to\infty\) and

$$\limsup_{n\to\infty}\frac{J(u_{n},v_{n})}{K_{n}}\leq0. $$

Define \(\tilde{u}_{n}=u_{n}\setminus K_{n}\), \(\tilde{v}_{n}=v_{n}\setminus K_{n}\), similar to the proof of the \((PS)\) condition of Theorem 1 again, we obtain that \((\tilde{u}_{n},\tilde{v}_{n})\) converges strongly to \(\pm(\mu_{1},\nu_{1})\) as \(n\to\infty\).

Assume that \((\tilde{u}_{n},\tilde{v}_{n})\) converges strongly to \((\mu_{1},\nu_{1})\) as \(n\to\infty\) (the case \((\tilde{u}_{n},\tilde{v}_{n})\) converges strongly to \(-(\mu_{1},\nu_{1})\) as \(n\to\infty\) may be treated similarly), from (14) we have

$$\begin{aligned} 0 \geq&\limsup_{n\to\infty}\frac{J(u_{n},v_{n})}{K_{n}} \\ \geq&\lim_{n\to\infty} \biggl(\int_{\Omega}h_{1}\tilde{u}_{n}+h_{2}\tilde{v}_{n} \,dx-\frac{1}{K_{n}}\int_{\Omega}F(x,u_{n},v_{n}) \,dx \biggr) \\ =&\int_{\Omega}(h_{1}\mu_{1}+h_{2} \nu_{1})\,dx-\int_{\Omega}\bigl(f^{++} \mu_{1}+g^{++}\nu_{1}\bigr)\,dx, \end{aligned}$$

which is a contradiction with \((LL)^{-}_{1}\). By Ekeland’s variational principle, Theorem 2 is proved. □