On existence of weak solutions for a p-Laplacian system at resonance

Abstract

This article shows the existence of weak solutions of a resonance problem for uniformly p-Laplacian system in a bounded domain in $R^N$. Our arguments are based on the Saddle Point Theorem (P.H.Rabinowitz) and rely on a generalization of the Landesman–Lazer type condition.

1 Introduction and preliminaries

Let $\Omega$ be a bounded domain in $R^N, (N\ge 3)$, with smooth boundary $\partial \Omega$. In the present paper we consider the existence of weak solutions of the following Dirichlet problem at resonance for p-Laplacian system:

$$\begin{aligned} \left\{ \begin{array}{lll} -\Delta _pu=&{}\lambda _1|u|^{\alpha -1}|v|^{\beta -1}v+f(x,u,v)-k_1(x) &{} \\ -\Delta _pv=&{}\lambda _1|u|^{\alpha -1}|v|^{\beta -1}u+g(x,u,v)-k_2(x) &{} \text { in } \Omega , \end{array} \right. \end{aligned}$$

(1.1)

where

$$\begin{aligned} p\ge 2, \alpha \ge 1, \beta \ge 1, \alpha +\beta =p \end{aligned}$$

(1.2)

and $f,g:\Omega \times R^2 \rightarrow R$ are Carathéodory functions which will be specified later.

$$\begin{aligned} k_i(x)\in L^{p'}(\Omega ), p'=\frac{p}{p-1}, k_i(x)>0, \text { for a.e } x\in \bar{\Omega }, i=1,2. \end{aligned}$$

$\lambda _1$ denotes the first eigenvalue of the problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta _pu=&{}\lambda |u|^{\alpha -1}|v|^{\beta -1}v \\ -\Delta _pv=&{}\lambda |u|^{\alpha -1}|v|^{\beta -1}u, \end{array}\right. } \end{aligned}$$

(1.3)

where $(u,v)\in E=W_0^{1,p}(\Omega )\times W_0^{1,p}(\Omega )$, $p\ge 2$, $\alpha \ge 1$, $\beta \ge 1$, $\alpha +\beta =p$.

It’s well-known that the principle eigenvalue $\lambda _1=\lambda _1(p)$ of (1.3) is obtained using the Ljusternick–Schnirelmann theory by minimizing the functional

$$\begin{aligned} J(u,v)=\frac{\alpha }{p}\int _{\Omega }|\nabla u|^p dx+\frac{\beta }{p}\int _{\Omega }|\nabla v|^p dx \end{aligned}$$

on the set:

$$\begin{aligned} S=\left\{ (u,v)\in E=W_0^{1,p}(\Omega )\times W_0^{1,p}(\Omega ): A(u,v)=1\right\} , \end{aligned}$$

where

$$\begin{aligned} A(u,v)=\int _{\Omega }|u|^{\alpha -1}|v|^{\beta -1}uvdx \end{aligned}$$

that is $\lambda _1=\lambda _1(p)$ can be variational characterized as

$$\begin{aligned} \lambda _1=\lambda _1(p)=\underset{A(u,v)>0}{inf}\frac{J(u,v)}{A(u,v)}. \end{aligned}$$

(1.4)

Moreover the eigenpair $(\varphi _1, \varphi _2)$ associated with $\lambda _1$ is componentwise positive and unique (up to multiplication by nonzero scalar) (see Theorem 2.2 in [3] and Remark 5.4 in [5]). As usual $W_0^{1,p}(\Omega )$ denotes Sobolev space which can be defined as the completion of $C_0^{\infty }(\Omega )$ under the norm:

$$\begin{aligned} ||u||_{W_0^{1,p}}=\left( \int _{\Omega }|\nabla u |^pdx\right) ^{\frac{1}{p}} \end{aligned}$$

and

$$\begin{aligned} \text {for }w=(u,v)\in E: ||w||_E=\left( ||u||^p_{W_0^{1,p}}+||v||^p_{W_0^{1,p}}\right) ^{\frac{1}{p}}. \end{aligned}$$

Observe that the existence of weak solutions of $(p,q)$-Laplacian systems at resonance in bounded domains with Dirichlet boundary condition, was first considered by Zographopoulos in [9]. Later in [4] Kandilakis and Magiropoulos have studied a quasilinear elliptic system with resonance part and nonlinear boundary condition in an unbounded domain by assuming the nonlinearities $f$ and $g$ depending only one variable $u$ or $v$. In [8] Zeng-Qi Ou and Chen Lei Tang have considered the same system as in [4] with Dirichlet condition in a bounded domain. In these the existence of weak solutions is obtained by critical point theory (the Minimum Principle or the Saddle Point Theorem ) under a Landesman–Lazer type condition.

In this paper by introducing a generalization of Landesman–Lazer type condition we shall prove an existence result for a p-Laplacian system on resonance in bounded domain with the nonlinearities $f$ and $g$ to be functions depending on both variables $u$ and $v$.

Our arguments are based on the Saddle Point Theorem (P.H.Rabinowitz) and generalization of the Landesman–Lazer type condition.

We have the following definition.

Definition 1.1

Function $w=(u,v)\in E$ is called a weak solution of the problem (1.1) if and only if, for all $\bar{w}=(\bar{u},\bar{v})\in E$

$$\begin{aligned}&\alpha \int _{\Omega }|\nabla u|^{p-2}\nabla u.\nabla \bar{u}dx+\beta \int _{\Omega }|\nabla v|^{p-2}\nabla v.\nabla \bar{v}dx\\&\quad -\lambda _1\int _{\Omega }(\alpha |u|^{\alpha -1}|v|^{\beta -1}v\bar{u}+\beta |u|^{\alpha -1}|v|^{\beta -1}u\bar{v})dx\\&\quad -\int _{\Omega }(\alpha f(x,u,v)\bar{u}+\beta g(x,u,v)\bar{v})dx+\int _{\Omega }(\alpha k_1(x)\bar{u}+\beta k_2(x)\bar{v})dx=0. \end{aligned}$$

We will use the following conditions

$(H_1)$

• (i) For a.e $x\in \Omega : f(x,.),g(x,.) \in C^1(R^2)$ and $f(x,0,0)=0, g(x,0,0)=0$.

• (ii) There exists function $\tau \in L^{p'}(\Omega ), p'=\frac{p}{p-1}$ such that:

  $$\begin{aligned} |f(x,s,t)|\le \tau (x), |g(x,s,t)|\le \tau (x), \text { for a.e } x\in \Omega , \forall (s,t)\in R^2. \end{aligned}$$

• (iii) For $(s,t)\in R^2$:

  $$\begin{aligned} \alpha \frac{\partial f(x,s,t)}{\partial t}=\beta \frac{\partial g(x,s,t)}{\partial s}\qquad \text { for a.e } x\in \Omega . \end{aligned}$$

  (1.5)

For $(u,v)\in R^2$, a.e $x\in \Omega$, define

$$\begin{aligned} H(x,u,v)=\frac{\alpha }{2}\int _0^u(f(x,s,v)+f(x,s,0))]ds+\frac{\beta }{2}\int _0^v(g(x,u,t)+g(x,0,t))dt. \end{aligned}$$

(1.6)

By hypotheses (1.5), from (1.6) with some simple computations we deduce that:

$$\begin{aligned} \frac{\partial H(x,s,t)}{\partial s}=\alpha f(x,s,t),\, \frac{\partial H(x,s,t)}{\partial t}=\beta g(x,s,t) \text {, for a.e } x\in \Omega , \forall (s,t)\in R^2. \end{aligned}$$

(1.7)

Now, for $i,j=1,2$ we define

$$\begin{aligned} F_{i}(x)&=\underset{\tau \rightarrow +\infty }{\text {lim}}\frac{1}{\tau }\int _0^\tau \left\{ f\left( x,(-1)^{1+i}y\varphi _1,(-1)^{1+i}\tau \varphi _2\right) +f\left( x,(-1)^{1+i}y\varphi _1,0\right) \right\} dy \nonumber \\ G_{j}(x)&=\underset{\tau \rightarrow +\infty }{\text {lim}}\frac{1}{\tau }\int _0^\tau \left\{ g\left( x,(-1)^{1+j}\tau \varphi _1,(-1)^{1+j}y\varphi _2\right) +g\left( x,0,(-1)^{1+j}y\varphi _2\right) \right\} dy \end{aligned}$$

(1.8)

and

$$\begin{aligned} \underset{{s \rightarrow +\infty }\above 0pt {t \rightarrow +\infty }}{\text {lim}}f(x,s,t)=f^{+\infty }(x),\qquad&\underset{{s \rightarrow +\infty }\above 0pt {t \rightarrow +\infty }}{\text {lim}}g(x,s,t)=g^{+\infty }(x) \\ \underset{{s \rightarrow -\infty }\above 0pt {t \rightarrow -\infty }}{\text {lim}}f(x,s,t)=f^{-\infty }(x),\qquad&\underset{{s \rightarrow -\infty }\above 0pt {t \rightarrow -\infty }}{\text {lim}}g(x,s,t)=g^{-\infty }(x). \end{aligned}$$

Assume that

$(H_2)$

• (i)
  $$\begin{aligned} f^{+\infty }(x)<k_1(x)<f^{-\infty }(x)&\nonumber \\ g^{+\infty }(x)<k_2(x)<g^{-\infty }(x)&\qquad \text { for a.e } x\in \Omega \end{aligned}$$

  (1.9)
• (ii)
  $$\begin{aligned}&\int _{\Omega }\left\{ \frac{1}{2}(\alpha F_2(x)\varphi _1(x)+\beta G_2(x)\varphi _2(x))-\frac{\alpha }{p}f^{-\infty }(x)\varphi _1(x)-\frac{\beta }{p}g^{-\infty }(x)\varphi _2(x)\right\} dx\nonumber \\&\quad <\left( 1-\frac{1}{p}\right) \int _{\Omega }(\alpha k_1(x)\varphi _1(x)+\beta k_2(x)\varphi _2(x))]dx\nonumber \\&\quad <\int _{\Omega }\left\{ \frac{1}{2}(\alpha F_1(x)\varphi _1(x)\!+\!\beta G_1(x)\varphi _2(x))-\frac{\alpha }{p}f^{+\infty }(x)\varphi _1(x)\!-\!\frac{\beta }{p}g^{+\infty }(x)\varphi _2(x)\right\} dx.\nonumber \\ \end{aligned}$$

  (1.10)

The main result of this paper can be described in the following theorem:

Theorem 1.1

Assuming conditions $(H_1)$, $(H_2)$ are fulfilled. Then the problem (1.1) has at least a nontrivial weak solution in $E$.

Proof of Theorem 1.1 is based on variational techniques and the Saddle Point Theorem (P.H.Rabinowitz).

Theorem 1.2

(Saddle Point Theorem, P.H.Rabinowitz in [6]) Let $E=X\oplus Y$ be a Banach space with $Y$ closed in $E$ and $dim X<\infty$. For $\varrho >0$ define

$$\begin{aligned} M:=\{u\in X:||u||\le \varrho \}\qquad \qquad M_0:=\{u\in X:||u||= \varrho \} \end{aligned}$$

Let $F\in C^1(E,R)$ be such that

$$\begin{aligned} b:=\underset{u\in Y}{\text {inf}}F(u)>a:=\underset{u\in M_0}{\text {max}}F(u) \end{aligned}$$

If $F$ satisfies the $(PS)_c$ condition with

$$\begin{aligned} c:=\underset{\gamma \in \Gamma }{\text {inf}}\underset{u\in M}{\text {max}} F(\gamma (u))\qquad \text {where } \Gamma :=\{\gamma \in C(M,E):\gamma |_{M_0}=I\}, \end{aligned}$$

then $c$ is a critical value of $F$.

2 Proof of the main result

We define the Euler–Lagrange functional associated to the problem (1.1) by

$$\begin{aligned} I(w)&= \frac{\alpha }{p}\int _{\Omega }|\nabla u|^pdx+\frac{\beta }{p}\int _{\Omega }|\nabla v|^pdx-\lambda _1\int _{\Omega }|u|^{\alpha -1}|v|^{\beta -1}u.vdx\nonumber \\&-\int _{\Omega }H(x,u,v)dx+\int _{\Omega }(\alpha k_1(x)u+\beta k_2(x)v)dx\nonumber \\&= J(w)+T(w), \quad \text { for } w=(u,v)\in E, \end{aligned}$$

(2.1)

where

$$\begin{aligned} J(w)&= \frac{\alpha }{p}\int _{\Omega }|\nabla u|^pdx+\frac{\beta }{p}\int _{\Omega }|\nabla v|^pdx. \end{aligned}$$

(2.2)

$$\begin{aligned} T(w)&= -\lambda _1\int _{\Omega }|u|^{\alpha -1}|v|^{\beta -1}u.vdx-\int _{\Omega }H(x,u,v)dx+\int _{\Omega }(\alpha k_1(x)u+\beta k_2(x)v)dx.\nonumber \\ \end{aligned}$$

(2.3)

We deduce that $I \in C^1(E).$

Remark 2.1

By similar arguments as those in the proof of Lemma 2.3 in [10] and Lemma 5 in [4], we infer that the functional $A :E\rightarrow R$ and the operator $B: E\rightarrow E^*$ given by, for any $(u,v), (\bar{u},\bar{v})\in E$

$$\begin{aligned} A(u,v)=\int _{\Omega }|u|^{\alpha -1}|v|^{\beta -1}u.vdx \end{aligned}$$

and

$$\begin{aligned} <B(u,v),(\bar{u},\bar{v})>=\int _{\Omega }|u|^{\alpha -1}|v|^{\beta -1}\bar{u}vdx+\int _{\Omega }|u|^{\alpha -1}|v|^{\beta -1}u\bar{v}dx, \end{aligned}$$

are compact.

Remark 2.2

Applying Theorem 1.6 in [6, p9] we deduce that the functional $J: E\rightarrow R$ given by (2.2) is weakly lower semicontinuous on $E$. Hence the functional $I=T+J$ is also weakly lower semicontinuous on $E$.

Proposition 2.1

Assuming the hypotheses $(H_1)$ and $(H_2)$ are fulfilled. The functional $I:E\rightarrow R$ given by (2.1) satisfies the $(PS)$ condition on $E$.

Proof

Let $\{w_m=(u_m,v_m)\}$ be a Palais–Smale sequence in $E$, i.e:

$$\begin{aligned} |I(w_m)|\le M, M \text { is positive constant} \end{aligned}$$

(2.4)

$$\begin{aligned} I'(w_m)\rightarrow 0 \text { in } E^* \text { as } m\rightarrow +\infty \end{aligned}$$

(2.5)

First, we shall prove that $\{w_m \}$ is bounded in $E$. We suppose by contradiction that $\{w_m\}$ is not bounded in $E$. Without loss of generality we assume that

$$\begin{aligned} ||w_m||_E\rightarrow +\infty \text { as } m\rightarrow +\infty . \end{aligned}$$

Let $\widehat{w}_m=\frac{w_m}{||w_m||_E}=(\widehat{u}_m,\widehat{v}_m)$ that is $\widehat{u}_m=\frac{u_m}{||w_m||_E}$ and $\widehat{v}_m=\frac{v_m}{||w_m||_E}$.

Thus $\widehat{w}_m$ is bounded in $E$. Then there exists a subsequence $\{\widehat{w}_{m_k}=(\widehat{u}_{m_k},\widehat{v}_{m_k})\}_k$ which converges weakly to $\widehat{w}=(\widehat{u},\widehat{v})$ in $E$. Since the embedding $W_0^{1,p}(\Omega )$ into $L^p(\Omega )$ is compact, the sequences $\{\widehat{u}_{m_k}\}$ and $\{\widehat{v}_{m_k}\}$ converge strongly to $\widehat{u}$ and $\widehat{v}$ in $L^p(\Omega )$ respectively.

From (2.4) we have

$$\begin{aligned}&\underset{k\rightarrow +\infty }{\text {lim}}\text {sup}\left\{ \frac{\alpha }{p}\int _{\Omega }|\nabla \widehat{u}_{m_k}|^pdx+\frac{\beta }{p}\int _{\Omega }|\nabla \widehat{v}_{m_k}|^pdx-\lambda _1\int _{\Omega }|\widehat{u}_{m_k}|^{\alpha -1}|\widehat{v}_{m_k}|^{\beta -1}\widehat{u}_{m_k}\widehat{v}_{m_k}dx\right. \nonumber \\&\quad \left. -\int _{\Omega }\frac{H(x,w_{m_k})}{||w_{m_k}||^p_E}dx+\int _{\Omega }\frac{\alpha k_1 \widehat{u}_{m_k}+\beta k_2 \widehat{v}_{m_k}}{||w_{m_k}||^{p-1}_E}dx\right\} \le 0. \end{aligned}$$

(2.6)

By hypotheses $(H_1)$, we deduce that

$$\begin{aligned} H(x,w_{mk})\!=\!\frac{\alpha }{2}\int _0^{u_{m_k}}(f(x,s,v_{mk})\!+\!f(x,s,0))ds\!+\!\frac{\beta }{2}\int _0^{v_{m_k}}(g(x,u_{mk},t)\!+\!g(x,0,t))dt. \end{aligned}$$

This implies that $|H(x,w_{mk})|\le c.\tau (x)(|u_{mk}|+|v_{mk}|)$, $c$ is positive constant.

Hence,

$$\begin{aligned} \left| \int _\Omega \frac{H(x,w_{mk})}{||w_{mk}||^p}\right| \le \frac{c}{||w_{mk}||^{p-1}_E}||\tau ||_{L^{p'}(\Omega )}\big (||\widehat{u}_{mk}||_{L^p(\Omega )}+||\widehat{v}_{mk}||_{L^p(\Omega )}\big ). \end{aligned}$$

Since $\widehat{u}_{m_k}$, $\widehat{v}_{m_k}$ converge strongly in $L^p(\Omega )$ then bounded in $L^p(\Omega )$, hence

$$\begin{aligned} \underset{k\rightarrow +\infty }{\text {lim}}\text {sup}\int _\Omega \frac{H(x,w_{mk})}{||w_{mk}||_E^p}=0 \end{aligned}$$

(2.7)

and

$$\begin{aligned} \underset{k\rightarrow +\infty }{\text {lim}}\int _{\Omega }\frac{\alpha k_1 \widehat{u}_{m_k}+\beta k_2 \widehat{v}_{m_k}}{||w_{m_k}||^{p-1}_E}dx=0. \end{aligned}$$

From the compactness of operator $A$ it follows that

$$\begin{aligned} \underset{k\rightarrow +\infty }{\text {lim}}\lambda _1\int _{\Omega }|\widehat{u}_{m_k}|^{\alpha -1}|\widehat{v}_{m_k}|^{\beta -1}\widehat{u}_{m_k}\widehat{v}_{m_k}dx=\lambda _1\int _{\Omega }|\widehat{u}|^{\alpha -1}|\widehat{v}|^{\beta -1}\widehat{u}.\widehat{v}dx. \end{aligned}$$

(2.8)

Using the weak lower semicontinuity of the functional $J$ and the variational characterization of $\lambda _1$ from (2.6) we get

$$\begin{aligned}&\lambda _1\int _{\Omega }|\widehat{u}|^{\alpha -1}|\widehat{v}|^{\beta -1}\widehat{u}.\widehat{v}dx\le \frac{\alpha }{p}\int _{\Omega }|\nabla \widehat{u}|^pdx+\frac{\beta }{p}\int _{\Omega }|\nabla \widehat{v}|^pdx \nonumber \\&\quad \le \underset{k\rightarrow +\infty }{\text {lim}} \text {inf} \left\{ \frac{\alpha }{p}\int _{\Omega }|\nabla \widehat{u}_{m_k}|^pdx+\frac{\beta }{p}\int _{\Omega }|\nabla \widehat{v}_{m_k}|^pdx\right\} \nonumber \\&\quad \le \underset{k\rightarrow +\infty }{\text {lim}} \text {sup} \left\{ \frac{\alpha }{p}\int _{\Omega }|\nabla \widehat{u}_{m_k}|^pdx+\frac{\beta }{p}\int _{\Omega }|\nabla \widehat{v}_{m_k}|^pdx\right\} \le \lambda _1\int _{\Omega }|\widehat{u}|^{\alpha -1}|\widehat{v}|^{\beta -1}\widehat{u}.\widehat{v}dx.\nonumber \\ \end{aligned}$$

(2.9)

Thus, theses inequalities are indeed equalities and we have

$$\begin{aligned} \underset{k\rightarrow +\infty }{\text {lim}} \left\{ \frac{\alpha }{p}\int _{\Omega }|\nabla \widehat{u}_{m_k}|^pdx+\frac{\beta }{p}\int _{\Omega }|\nabla \widehat{v}_{m_k}|^pdx\right\}&= \frac{\alpha }{p}\int _{\Omega }|\nabla \widehat{u}|^pdx+\frac{\beta }{p}\int _{\Omega }|\nabla \widehat{v}|^pdx\nonumber \\&= \lambda _1\int _{\Omega }|\widehat{u}|^{\alpha -1}|\widehat{v}|^{\beta -1}\widehat{u}.\widehat{v}dx. \end{aligned}$$

(2.10)

We shall prove that $\widehat{u}\not = 0$ and $\widehat{v}\not =0$.

By contradiction suppose that $\widehat{u}=0$, thus $\widehat{u}_{m_k}\rightarrow 0$ in $L^p(\Omega )$ as $k\rightarrow +\infty$. We have

$$\begin{aligned} |A(\widehat{u}_{m_k},\widehat{v}_{m_k})|&= \left| \int _{\Omega }|\widehat{u}_{m_k}|^{\alpha -1}|\widehat{v}_{m_k}|^{\beta -1}\widehat{u}_{m_k}\widehat{v}_{m_k}dx\right| \\&\le ||\widehat{u}_{m_k}||^{\alpha }_{L^p(\Omega )}.||\widehat{v}_{m_k}||^{\beta }_{L^p(\Omega )}. \end{aligned}$$

Since $||\widehat{u}_{m_k}||_{L^p(\Omega )}\rightarrow 0$, letting $k\rightarrow +\infty$ shows that

$$\begin{aligned} \underset{k\rightarrow +\infty }{\text {lim}}A(\widehat{u}_{m_k},\widehat{v}_{m_k})=0. \end{aligned}$$

(2.11)

From (2.6) taking $\underset{k\rightarrow +\infty }{\text {lim}}\text {sup}$ with (2.7) and (2.10) we arrive at

$$\begin{aligned} \underset{k\rightarrow +\infty }{\text {lim}}\text {sup}\left\{ \frac{\alpha }{p}\int _{\Omega }|\nabla \widehat{u}_{m_k}|^pdx+\frac{\beta }{p}\int _{\Omega }|\nabla \widehat{v}_{m_k}|^pdx\right\} =0. \end{aligned}$$

(2.12)

On the other hand, since $||\widehat{w}_{m_k}||_E=1$ and

$$\begin{aligned} \frac{\alpha }{p}\int _{\Omega }|\nabla \widehat{u}_{m_k}|^pdx+\frac{\beta }{p}\int _{\Omega }|\nabla \widehat{v}_{m_k}|^pdx\ge \text {min}\left( \frac{\alpha }{p},\frac{\beta }{p}\right) .||\widehat{w}_{m_k}||_E=\text {min}\left( \frac{\alpha }{p},\frac{\beta }{p}\right) >0 \end{aligned}$$

which contradicts (2.11). Thus $\widehat{u}\not =0$. Similary we have $\widehat{v}\not =0$.

By again the definition of $\lambda _1$ from (2.10) we deduce that

$$\begin{aligned} \widehat{w}=(\widehat{u},\widehat{v})=(\varphi _1,\varphi _2) \text { or } \widehat{w}=(\widehat{u},\widehat{v})=(-\varphi _1,-\varphi _2), \end{aligned}$$

where $({\varphi _1},{\varphi _2})$ is eigenpair associated with $\lambda _1$ of the problem (1.3).

Next, we shall consider following two cases:

Firstly, assume that $\widehat{u}_{m_k}\rightarrow {\varphi _1},\widehat{v}_{m_k}\rightarrow {\varphi _2}$ in $L^p(\Omega )$ as $k\rightarrow +\infty$.

From (2.4) we have

$$\begin{aligned} -M&\le -\frac{\alpha }{p}\int _{\Omega }|\nabla u_{m_k}|^pdx-\frac{\beta }{p}\int _{\Omega }|\nabla v_{m_k}|^pdx+\lambda _1\int _{\Omega }|u_{m_k}|^{\alpha -1}|v_{m_k}|^{\beta -1}u_{m_k}v_{m_k}dx\nonumber \\&+\int _{\Omega }H(x,w_{m_k})dx-\int _{\Omega }(\alpha k_1u_{m_k}+\beta k_2v_{m_k})dx\le M. \end{aligned}$$

(2.13)

Moreover, from (2.5) there exists the sequence $\epsilon _k$, $\epsilon _k\rightarrow 0^+$, $k\rightarrow +\infty$ such that

$$\begin{aligned} |<I'(w_{m_k}),\left( \frac{u_{m_k}}{p},\frac{v_{m_k}}{p}\right) >| \le \epsilon _k.\frac{1}{p}||w_m||_E. \end{aligned}$$

This implies

$$\begin{aligned}&-\epsilon _{k}.\frac{1}{p}||w_{m_k}||_E\le \alpha \int _{\Omega }|\nabla u_{m_k}|^{p-2}\nabla u_{m_k}\nabla \left( \frac{u_{m_k}}{p}\right) dx+\beta \int _{\Omega }|\nabla v_{m_k}|^{p-2}\nabla v_{m_k}\nabla \left( \frac{v_{m_k}}{p}\right) dx\nonumber \\&\qquad -\lambda _1\int _{\Omega }\left( \alpha |u_{m_k}|^{\alpha -1}|v_{m_k}|^{\beta -1}v_{m_k}\left( \frac{u_{m_k}}{p}\right) +\beta |u_{m_k}|^{\alpha -1}|v_{m_k}|^{\beta -1}u_{m_k}\left( \frac{v_{m_k}}{p}\right) \right) dx\nonumber \\&\qquad -\int _{\Omega }\left( \alpha f\big (x,w_{m_k}\big )\frac{u_{m_k}}{p}+\beta g\big (x,w_{m_k}\big )\frac{v_{m_k}}{p}\right) dx+\int _{\Omega }\left( \alpha k_1\frac{u_{m_k}}{p}+\beta k_2\frac{v_{m_k}}{p}\right) dx \nonumber \\&\quad \le \epsilon _{k}.\frac{1}{p}||w_{m_k}||_E. \end{aligned}$$

Remark that $\alpha +\beta =p$, we get

$$\begin{aligned}&-\epsilon _{k}.\frac{1}{p}||w_{m_k}||_E\le \frac{\alpha }{p} \int _{\Omega }|\nabla u_{m_k}|^pdx+\frac{\beta }{p} \int _{\Omega }|\nabla v_{m_k}|^pdx\nonumber \\&\quad -\lambda _1\int _{\Omega }\big (\alpha |u_{m_k}|^{\alpha -1}|v_{m_k}|^{\beta -1}u_{m_k}v_{m_k}\big )dx\!-\!\int _{\Omega }\left( \alpha f\big (x,w_{m_k}\big )\frac{u_{m_k}}{p}+\beta g\big (x,w_{m_k}\big )\frac{v_{m_k}}{p}\right) dx\nonumber \\&\quad +\int _{\Omega }\left( \frac{\alpha }{p} k_1 u_{m_k}+\frac{\beta }{p} k_2v_{m_k}\right) dx\le \epsilon _{k}.\frac{1}{p}||w_{m_k}||_E. \end{aligned}$$

(2.14)

Hence, summing (2.13), (2.14) we obtain

$$\begin{aligned}&-M-\frac{\epsilon _k}{p}||w_{m_k}||_E\le \int _{\Omega } \left( H\big (x,w_{m_k}\big )-\left( \frac{\alpha }{p}f\big (x,w_{m_k}\big )u_{m_k}+\frac{\beta }{p}g\big (x,w_{m_k}\big )v_{m_k}\right) \right) dx \nonumber \\&\quad -\int _{\Omega }\left( \alpha \left( 1-\frac{1}{p}\right) k_1 u_{m_k}+\beta \left( 1-\frac{1}{p}\right) k_2 v_{m_k}\right) dx\le M+\frac{\epsilon _k}{p}||w_{m_k}||_E. \end{aligned}$$

(2.15)

After dividing (2.15) by $||w_{m_k}||_E$, letting $\underset{k\rightarrow +\infty }{\text {lim}}\text {sup}$ we deduce that

$$\begin{aligned}&\underset{k\rightarrow +\infty }{\text {lim}}\text {sup}\int _{\Omega }\left\{ \frac{H(x,w_{m_k})}{||w_{m_k}||_E}-\frac{\alpha }{p}f(x,w_{m_k})\widehat{u}_{m_k}-\frac{\beta }{p}g(x,w_{m_k})\widehat{v}_{m_k}\right\} dx\nonumber \\&\quad =\left( 1-\frac{1}{p}\right) \int _{\Omega }(\alpha k_1 \varphi _1+\beta k_2\varphi _2)dx. \end{aligned}$$

(2.16)

We remark that, from (1.6) by some standard computations we get

$$\begin{aligned} \underset{k\rightarrow +\infty }{\text {lim}}\text {sup}\int _{\Omega }\frac{H(x,w_{m_k})}{||w_{m_k}||_E}dx=\frac{1}{2}\int _{\Omega }(\alpha F_1\varphi _1+\beta G_1\varphi _2)dx, \end{aligned}$$

where $F_1(x), G_1(x)$ are given by (1.8).

Letting $\underset{k\rightarrow +\infty }{\text {lim}}\text {sup}$ (2.16) we obtain

$$\begin{aligned}&\int _{\Omega }\left\{ \frac{1}{2}(\alpha F_1\varphi _1+\beta G_1\varphi _2)-\frac{\alpha }{p}f^{+\infty }\varphi _1-\frac{\beta }{p}g^{+\infty }\varphi _2\right\} dx\\&\quad =\left( 1-\frac{1}{p}\right) \int _{\Omega }(\alpha k_1 \varphi _1+\beta k_2\varphi _2)dx, \end{aligned}$$

which contradicts $(H_2(ii))$.

Similarly, in the case when $\widehat{u}_{m_k}\rightarrow {-\varphi _1}, \widehat{v}_{m_k}\rightarrow {-\varphi _2}$, in $L^p(\Omega )$ as $k\rightarrow +\infty$, by similar computations, we also have

$$\begin{aligned}&\int _{\Omega }\left\{ \frac{1}{2}(\alpha F_2\varphi _1+\beta G_2\varphi _2)-\frac{\alpha }{p}f^{-\infty }\varphi _1-\frac{\beta }{p}g^{-\infty }\varphi _2\right\} dx\\&\quad =\left( 1-\frac{1}{p}\right) \int _{\Omega }(\alpha k_1 \varphi _1+\beta k_2\varphi _2)dx, \end{aligned}$$

where $F_2(x), G_2(x)$ are given by (1.8), which contradicts $(H_2(ii))$.

This implies that the $(PS)$ sequence $\{w_m\}$ is bounded in $E$. Then there exists a subsequence $w_{m_k}$ which converges weakly to $w_0=(u_0,v_0)\in E.$

We shall prove that $w_{m_k}$ converges strongly to $w_0=(u_0,v_0)\in E.$

Indeed, since $w_{m_k}\rightharpoonup w_0=(u_0,v_0)$ in $E$ and the embedding $W^{1,p}_0\times W^{1,p}_0\hookrightarrow L^p(\Omega )\times L^p(\Omega )$ is compact, the subsequences $u_{m_k}$, $v_{m_k}$ converge strongly to $u_0,v_0$ in $L^p$ respectively. We have

$$\begin{aligned}&|T'(w_{m_k},(w_{m_k}-w_0))|\le \lambda _1\left\{ \int _{\Omega }\alpha |u_{m_k}|^{\alpha -1}|v_{m_k}|^\beta |u_{m_k}-u_0|dx\right. \nonumber \\&\qquad +\left. \int _{\Omega }\beta |u_{m_k}|^{\alpha }|v_{m_k}|^{\beta -1}|v_{m_k}-v_0|dx\right\} +\int _{\Omega }\left\{ \alpha |f(x,w_{m_k})||u_{m_k}-u_0|\right. \nonumber \\&\qquad +\left. \beta |g(x,w_{m_k})||v_{m_k}-v_0|\right\} dx+\int _{\Omega }\left\{ \alpha k_1(x)|u_{m_k}-u_0|+\beta k_2(x)|v_{m_k}-v_0|\right\} dx \nonumber \\&\quad \le \lambda _1\left\{ \alpha ||u_{m_k}||^{\alpha -1}_{L^p}||v_{m_k}||^{\beta }_{L^p}||u_{m_k}-u_0||_{L^p}+\beta ||u_{m_k}||^{\alpha }_{L^p}||v_{m_k}||^{\beta -1}_{L^p}||v_{m_k}-v_0||_{L^p}\right\} \nonumber \\&\qquad +\,||\tau ||_{L^{p'}}(\alpha ||u_{m_k}-u_0||_{L^p}+\beta ||v_{m_k}-v_0||_{L^p})\nonumber \\&\qquad +\,\alpha ||k_1||_{L^{p'}}||u_{m_k}-u_0||_{L^p} +\beta ||k_2||_{L^{p'}}||u_{m_k}-u_0||_{L^p}. \end{aligned}$$

(2.17)

Letting $k\rightarrow +\infty$ and remark that $||u_{m_k}-u_0||_{L^p}\rightarrow 0$, $||v_{m_k}-v_0||_{L^p}\rightarrow 0$. We obtain

$$\begin{aligned} \underset{k\rightarrow +\infty }{\text {lim}}<T'(w_{m_k}),(w_{m_k}-w_0)>\,=\,0. \end{aligned}$$

Moreover,

$$\begin{aligned} \underset{k\rightarrow +\infty }{\text {lim}}(J'(w_{m_k}),(w_{m_k}-w_0)) =\underset{k\rightarrow +\infty }{\text {lim}}\left\{ (I'(w_{m_k}),(w_{m_k}-w_0))-(T'(w_{m_k}),(w_{m_k}-w_0))\right\} . \end{aligned}$$

We have

$$\begin{aligned} \underset{k\rightarrow +\infty }{\text {lim}}(J'(w_{m_k}),(w_{m_k}-w_0))=0 \end{aligned}$$

i.e

$$\begin{aligned} (J'(w_{m_k}),(w_{m_k}-w_0))&= \alpha \int _{\Omega }|\nabla u_{m_k}|^{p-2}|\nabla u_{m_k}|\nabla (u_{m_k}-u_0)dx\nonumber \\&+\,\beta \int _{\Omega }|\nabla v_{m_k}|^{p-2}|\nabla v_{m_k}|\nabla (v_{m_k}-v_0)dx\rightarrow 0 \qquad \text { as } k\rightarrow +\infty .\nonumber \\ \end{aligned}$$

(2.18)

Since $w_{m_k}\rightharpoonup w_0$ in $E$ and $J'(w_0)\in E^*$,$(J'(w_0),(w_{m}-w_0))\rightarrow 0$ as $k\rightarrow +\infty$.

That is

$$\begin{aligned} (J'(w_{0}),(w_{m_k}-w_0))&= \alpha \int _{\Omega }|\nabla u_{0}|^{p-2}|\nabla u_{0}|\nabla (u_{m_k}-u_0)dx\nonumber \\&+\,\beta \int _{\Omega }|\nabla v_{0}|^{p-2}|\nabla v_{0}|\nabla (v_{m_k}-v_0)dx \rightarrow 0, \qquad \text { as } k\rightarrow +\infty .\nonumber \\ \end{aligned}$$

(2.19)

Using the well-know inequality:

$$\begin{aligned} (|s|^{r-2}s-|\bar{s}|^{r-2})(s-\bar{s})\ge c_r |s-\bar{s}|^{r}, \end{aligned}$$

for $s,\bar{s} \in R^N$, $r\ge 2$, we deduce that

$$\begin{aligned}&<J'(w_{m_k})-J'(w_0),(w_{m_k}-w_0)>\\&\quad =\alpha \int _{\Omega }(|\nabla u_{m_k}|^{p-2}\nabla u_{m_k}-|\nabla u_{0}|^{p-2}\nabla u_{0})\nabla (u_{m_k}-u_0)dx\\&\qquad +\,\beta \int _{\Omega }(|\nabla v_{m_k}|^{p-2}\nabla v_{m_k}-|\nabla v_{0}|^{p-2}\nabla v_{0})\nabla (v_{m_k}-v_0)dx \\&\quad \ge c_1||u_{m_k}-u_0||_{W^{1,p}_0}+c_2||v_{m_k}-v_0||_{W^{1,p}_0}. \end{aligned}$$

From (2.18), (2.19) it follows that the left-hand side of this inequality converges to zero as $k\rightarrow +\infty$. Then we arrive at $u_{m_k}\rightarrow u_0$, $v_{m_k}\rightarrow v_0$ as $k\rightarrow +\infty$ in $W_0^{1,p}(\Omega )$.

Hence, we deduce that $\{w_{m_k}\}$ converges strongly to $w_0$ in $E$.

Therefore, the functional $I$ satisfies the Palais$-$Smale condition in $E$.

The proof of the Proposition 2.1 is complete.

Splitting $E$ as the direct sum of $X,Y$: $E=X\oplus Y$ where

$$\begin{aligned} X&= L(\varphi )=\{t\varphi =t(\varphi _1,\varphi _2),\quad t\in R\}\\ Y&= \left\{ w=(u,v)\in E:\int _{\Omega }(u\varphi _1^{\alpha -1}\varphi _2^{\beta }+v\varphi _1^{\alpha }\varphi _2^{\beta -1})dx=0\right\} , \end{aligned}$$

where $\varphi =(\varphi _1,\varphi _2)$ is a nomarlized eigenpair associated with the eigenvalue $\lambda _1$ of the problem (1.3)

$$\begin{aligned} ||(\varphi _1,\varphi _2)||=\left( \int _{\Omega }|\nabla \varphi _1|^pdx+\int _{\Omega }|\nabla \varphi _2|^pdx\right) ^{\frac{1}{p}}=1. \end{aligned}$$

Since $w=(u,v)\in E$, $w=t(\varphi _1,\varphi _2)+w_0$, $w_0=(u_0,v_0)\in Y$.

$$\begin{aligned} u&= t\varphi _1+u_0 \end{aligned}$$

(2.20)

$$\begin{aligned} v&= t\varphi _2+v_0 \end{aligned}$$

(2.21)

Multiplying the equations in (2.20), (2.21) by $\varphi _1^{\alpha -1}\varphi _2^{\beta }\lambda _1$ and $\varphi _1^{\alpha }\varphi _2^{\beta -1}\lambda _1$ respectively, we have

$$\begin{aligned} \lambda _1 u\varphi _1^{\alpha -1}\varphi _2^{\beta }&= \lambda _1 t\varphi _1^{\alpha }\varphi _2^{\beta }+\lambda _1 u_0\varphi _1^{\alpha -1}\varphi _2^{\beta }. \end{aligned}$$

(2.22)

$$\begin{aligned} \lambda _1 v\varphi _1^{\alpha }\varphi _2^{\beta -1}&= \lambda _1 t\varphi _1^{\alpha }\varphi _2^{\beta }+\lambda _1 v_0\varphi _1^{\alpha }\varphi _2^{\beta -1}. \end{aligned}$$

(2.23)

We remark that

$$\begin{aligned} -\Delta _p\varphi _1=-\text {div} (|\nabla \varphi _1|^{p-2}\nabla \varphi _1)=\lambda _1\varphi _1^{\alpha -1}\varphi _2^{\beta }. \end{aligned}$$

From (2.22) we have $\lambda _1 u \varphi _1^{\alpha -1}\varphi _2^{\beta }=t(-\text {div} (|\nabla \varphi _1|^{p-2}\nabla \varphi _1))\varphi _1+\lambda _1 u_0\varphi _1^{\alpha -1}\varphi _2^{\beta }$.

By integrating both sides of (2.22), we obtain that

$$\begin{aligned} \lambda _1\int _{\Omega } u \varphi _1^{\alpha -1}\varphi _2^{\beta }dx&= t\int _{\Omega }\left( -\text {div} (|\nabla \varphi _1|^{p-2}\nabla \varphi _1)\right) \varphi _1dx+\lambda _1\int _{\Omega } u_0\varphi _1^{\alpha -1}\varphi _2^{\beta }dx \nonumber \\&= t\int _{\Omega }|\nabla \varphi _1|^pdx+\lambda _1\int _{\Omega }u_0\varphi _1^{\alpha -1}\varphi _2^{\beta }dx. \end{aligned}$$

(2.24)

Similary, from (2.23) we also have

$$\begin{aligned} \lambda _1\int _{\Omega } v\varphi _1^{\alpha }\varphi _2^{\beta -1}dx=t\int _{\Omega }|\nabla \varphi _2|^pdx+\lambda _1\int _{\Omega }v_0\varphi _1^{\alpha }\varphi _2^{\beta -1}dx. \end{aligned}$$

(2.25)

Hence combining (2.24) and (2.25) we obtain

$$\begin{aligned} \lambda _1\int _{\Omega }\left( u \varphi _1^{\alpha -1}\varphi _2^{\beta }+v\varphi _1^{\alpha }\varphi _2^{\beta -1}\right) dx&= t\int _{\Omega }|\nabla \varphi _1|^pdx+\lambda _1\int _{\Omega }u_0\varphi _1^{\alpha -1}\varphi _2^{\beta }dx\\&+\, t\int _{\Omega }|\nabla \varphi _2|^pdx+\lambda _1\int _{\Omega }v_0\varphi _1^{\alpha }\varphi _2^{\beta -1}dx. \end{aligned}$$

Since $(u_0,v_0)\in Y$, we have

$$\begin{aligned} \int _{\Omega }\left( u_0 \varphi _1^{\alpha -1}\varphi _2^{\beta }+v_0\varphi _1^{\alpha }\varphi _2^{\beta -1}\right) dx =0. \end{aligned}$$

Thus, for any $w\in E$ such that $w=t\varphi +w_0$, $w_0\in Y$ we get

$$\begin{aligned} t=\frac{\lambda _1\int _{\Omega }\left( u \varphi _1^{\alpha -1}\varphi _2^{\beta }+v\varphi _1^{\alpha }\varphi _2^{\beta -1}\right) dx}{\int _{\Omega }|\nabla \varphi _1|^pdx+\int _{\Omega }|\nabla \varphi _2|^pdx}=\lambda _1\int _{\Omega }\left( u \varphi _1^{\alpha -1}\varphi _2^{\beta }+v\varphi _1^{\alpha }\varphi _2^{\beta -1}\right) dx. \end{aligned}$$

(2.26)

Moreover, if $w=t\varphi +\tilde{w}$ where $t$ is defined in (2.26) then $\tilde{w}\in Y.$

Therefore, $E=X\oplus Y.$

Lemma 2.1

Exists $\bar{\lambda }>\lambda _1$ such that

$$\begin{aligned} \frac{\alpha }{p}\int _{\Omega }|\nabla u|^pdx+\frac{\beta }{p}\int _{\Omega }|\nabla v|^pdx\ge \bar{\lambda }\int _{\Omega }|u|^{\alpha -1}|v|^{\beta -1}uvdx\text {, }\forall w=(u,v)\in Y. \end{aligned}$$

Proof

Let $\lambda =\text {inf}\{ \frac{\alpha }{p}\int _{\Omega }|\nabla u|^pdx+\frac{\beta }{p}\int _{\Omega }|\nabla v|^pdx: (u,v)\in Y, \int _{\Omega }|u|^{\alpha -1}|v|^{\beta -1}uvdx=1 \}$.

We shall prove that this value is attained in $Y$.

Let $w_m=(u_m,v_m)\in Y$ be a minimizing sequence i.e

$$\begin{aligned} \int _{\Omega }|u_m|^{\alpha -1}|v_m|^{\beta -1}u_mv_mdx=1, \quad \text {for } m=1,2,... \end{aligned}$$

and

$$\begin{aligned} \underset{m\rightarrow +\infty }{\text {lim}}\frac{\alpha }{p}\int _{\Omega }|\nabla u_m|^pdx+\frac{\beta }{p}\int _{\Omega }|\nabla v_m|^pdx=\lambda . \end{aligned}$$

This implies that $\{w_m\}$ is bounded in $E$. Hence there exists a subsequence $\{w_{m_k}\}$ of $\{w_m\}$ which weakly converges to $w_0=(u_0,v_0)\in E$ and the compactness of the embedding $W_0^{1,p}(\Omega )$ into $L^p(\Omega )$ implies that the subsequences $\{u_{m_k}\}$ and $\{v_{m_k}\}$ converge strongly to $u_0$ and $v_0$ respectively in $L^p(\Omega )$.

Observe further that with $\alpha +\beta =p$

$$\begin{aligned}&\int _{\Omega }\left( (u_{m_k}-u_0)\varphi _1^{\alpha -1}\varphi _2^{\beta }+(v_{m_k}-v_0)\varphi _1^{\alpha }\varphi _2^{\beta -1}\right) dx \\&\quad \le ||u_{m_k}-u_0||_{L^p}||\varphi _1||^{\alpha -1}_{L^p}|\varphi _2||^{\beta }_{L^p}+||v_{m_k}-v_0||_{L^p}||\varphi _1||^{\alpha }_{L^p}|\varphi _2||^{\beta -1}_{L^p}. \end{aligned}$$

Since $||u_{m_k}-u_0||_{L^p(\Omega )}\rightarrow 0$, $||v_{m_k}-v_0||_{L^p(\Omega )}\rightarrow 0$ as $k\rightarrow +\infty$, we deduce that

$$\begin{aligned} \underset{k\rightarrow +\infty }{\text {lim}}\int _{\Omega }\left( u_{m_k}\varphi _1^{\alpha -1}\varphi _2^{\beta }+v_{m_k}\varphi _1^{\alpha }\varphi _2^{\beta -1}\right) dx= \int _{\Omega }\left( u_{0}\varphi _1^{\alpha -1}\varphi _2^{\beta }+v_{0}\varphi _1^{\alpha }\varphi _2^{\beta -1}\right) dx. \end{aligned}$$

From this it follows that

$$\begin{aligned} \int _{\Omega }\left( u_{0}\varphi _1^{\alpha -1}\varphi _2^{\beta }+v_{0}\varphi _1^{\alpha }\varphi _2^{\beta -1}\right) dx =0, \end{aligned}$$

hence $(u_0,v_0)\in Y$.

On the other hand, by the continuity of the operator $A$

$$\begin{aligned} \underset{k\rightarrow +\infty }{\text {lim}}\int _{\Omega }|u_{m_k}|^{\alpha -1}|v_{m_k}|^{\beta -1}u_{m_k}v_{m_k}dx=\int _{\Omega }|u_{0}|^{\alpha -1}|v_{0}|^{\beta -1}u_{0}v_{0}dx. \end{aligned}$$

This implies

$$\begin{aligned} \int _{\Omega }|u_{0}|^{\alpha -1}|v_{0}|^{\beta -1}u_{0}v_{0}dx =1. \end{aligned}$$

So $u_0 \not = 0$ and $v_0 \not = 0$.

Moreover, since the functional $J$ given by (2.2) is lower weakly semicontinuous, we obtain

$$\begin{aligned} \lambda \le J(u_0,v_0)&= \frac{\alpha }{p}\int _{\Omega }|\nabla u_{m_k}|^pdx+\frac{\beta }{p}\int _{\Omega }|\nabla v_{m_k}|^pdx\\&\le \underset{m\rightarrow +\infty }{\text {lim}} inf \left\{ \frac{\alpha }{p}\int _{\Omega }|\nabla u_{m_k}|^pdx+\frac{\beta }{p}\int _{\Omega }|\nabla v_{m_k}|^pdx\right\} =\lambda , \end{aligned}$$

hence

$$\begin{aligned} \lambda =J(u_0,v_0)=\frac{\alpha }{p}\int _{\Omega }|\nabla u_0|^pdx+\frac{\beta }{p}\int _{\Omega }|\nabla v_0|^pdx. \end{aligned}$$

It means that $\lambda$ is attained at $w_0$.

Our goal is to show that $\lambda >\lambda _1$.

By the variational characterization of $\lambda _1$, it is clear that: $\lambda \ge \lambda _1$.

If $\lambda =\lambda _1$, by simplicity of $\lambda _1$ there exists $t\in R$ such that $w_0=(u_0,v_0)=t(\varphi _1,\varphi _2)$. Since $w_0=(u_0,v_0)\in Y$

$$\begin{aligned} 0=\int _{\Omega }\left( t\varphi _1 \varphi _1^{\alpha -1}\varphi _2^{\beta }+t\varphi _2 \varphi _1^{\alpha }\varphi _2^{\beta -1}\right) dx=t\int _{\Omega }\varphi _1^{\alpha }\varphi _2^{\beta }dx. \end{aligned}$$

This contradicts the fact that

$$\begin{aligned} 1= \int _{\Omega }|u_{0}|^{\alpha -1}|v_{0}|^{\beta -1}u_{0}v_{0}dx =t\int _{\Omega }\varphi _1^{\alpha }\varphi _2^{\beta }dx. \end{aligned}$$

Thus, there exists $\bar{\lambda }$ such that: $\bar{\lambda }>\lambda _1$ and the proof of proposition is complete. $\square$

Proposition 2.2

The functional $I$ given by (2.1) is coercive on $Y$ provided hypotheses $(H_1)$ and $(H_2)$ hold.

Proof

Observe that by Holder inequality, Lemma 2.1, hypotheses $(H_1)$, $(H_2)$, we have

$$\begin{aligned}&|I(w)|=|\frac{\alpha }{p}\int _{\Omega }|\nabla u|^pdx+\frac{\beta }{p}\int _{\Omega }|\nabla v|^pdx-\lambda _1\int _{\Omega }|u|^{\alpha -1}|v|^{\beta -1}uvdx\\&\qquad - \int _{\Omega }H(x,u,v)dx+\int _{\Omega }(\alpha k_1 u+\beta k_2 v)dx|\\&\quad \ge | \text {min}\left( \frac{\alpha }{p};\frac{\beta }{p}\right) ||w||_E^p -\frac{\lambda _1}{\bar{\lambda }}\left( \frac{\alpha }{p}\int _{\Omega }|\nabla u|^pdx+\frac{\beta }{p}\int _{\Omega }|\nabla v|^pdx\right) \\&\qquad -\int _{\Omega }\tau (x)(|u|+|v|)dx-\alpha ||k_1||_{L^{p'}}||u||_{L^p}-\beta ||k_2||_{L^{p'}}||v||_{L^p}|\\&\quad \ge |\left( 1-\frac{\lambda _1}{\bar{\lambda }}\right) \text {min}\left( \frac{\alpha }{p};\frac{\beta }{p}\right) ||w||_E^p-(||\tau ||_{L^{p'}}\\&\qquad +\,\alpha ||k_1||_{L^{p'}})||u||_{L^p}-(||\tau ||_{L^{p'}}+\beta ||k_2||_{L^{p'}})||v||_{L^p}|\\&\quad \ge |\left( 1\!-\!\frac{\lambda _1}{\bar{\lambda }}\right) \text {min}\left( \frac{\alpha }{p};\frac{\beta }{p}\right) ||w||_E^p\!-\!\text {max}\left\{ (||\tau ||_{L^{p'}}\!+\!\alpha ||k_1||_{L^{p'}}),(||\tau ||_{L^{p'}}\!+\!\beta ||k_2||_{L^{p'}})\!\right\} \!.\\&\qquad .c(||u||_{W_0^{1,p}}+||v||_{W_0^{1,p}})|. \end{aligned}$$

Since $||w_E||\rightarrow +\infty$ and $\left( 1-\frac{\lambda _1}{\bar{\lambda }}\right) >0$, $p\ge 2$, we obtain $I(w)\rightarrow +\infty$.

Thus the functional $I$ given by (2.1) is coercive on $Y$ and Proposition 2.2 is proved. $\square$

From Proposition 2.1 the functional $I$ is coercive on $Y$, so that

$$\begin{aligned} B_Y=\underset{w\in Y}{\text {min}}I(w)>-\infty . \end{aligned}$$

On the other hand, for every $t\in R$ we have

$$\begin{aligned} \frac{\alpha }{p}\int _{\Omega }|\nabla (t\varphi _1)|^pdx+\frac{\beta }{p}\int _{\Omega }|\nabla (t\varphi _2)|^pdx-\lambda _1\int _{\Omega }|t\varphi _1|^{\alpha -1}|t\varphi _2|^{\beta -1}(t\varphi _1)(t\varphi _2)dx=0 \end{aligned}$$

as follows from the definition of $\lambda _1$ and $\varphi$. Thus,

$$\begin{aligned} I(t\varphi )&= t\int _{\Omega }(\alpha k_1\varphi _1+\beta k_2 \varphi _2)dx-\int _{\Omega }H(x,t\varphi )dx\\&= t\int _{\Omega }\Big ((\alpha k_1\varphi _1+\beta k_2 \varphi _2)-\frac{H(x,t\varphi )}{t}\Big )dx. \end{aligned}$$

Remark that

$$\begin{aligned} \frac{H(x,t\varphi )}{t}&= \frac{1}{t}\left\{ \frac{\alpha }{2}\int _0^{t\varphi _1}(f(x,s,t\varphi _2)+f(x,s,0))ds\right. \\&\qquad +\left. \frac{\beta }{2}\int _0^{t\varphi _2}(g(x,t\varphi _1,\tau )+g(x,0,\tau ))d\tau \right\} \\&= \frac{1}{t}\left\{ \frac{\alpha }{2}\int _0^{t}((f(x,y\varphi _1,t\varphi _2)+f(x,y\varphi _1,0))dy)\varphi _1\right. \\&\qquad +\left. \frac{\beta }{2}\int _0^{t}((g(x,t\varphi _1,y\varphi _2)+g(x,0,y\varphi _2))dy)\varphi _2\right\} . \end{aligned}$$

Hence,

$$\begin{aligned} \underset{t\rightarrow +\infty }{\text {lim}}\frac{H(x,t\varphi )}{t}=\frac{1}{2}(\alpha F_1(x)\varphi _1+\beta G_1(x)\varphi _2). \end{aligned}$$

Therefore,

$$\begin{aligned}&\underset{t\rightarrow +\infty }{\text {lim}}t\int _{\Omega }\Big ((\alpha k_1\varphi _1+\beta k_2 \varphi _2)-\frac{H(x,t\varphi )}{t}\Big )dx\\&\quad =\underset{t\rightarrow +\infty }{\text {lim}}t\int _{\Omega }\left\{ (\alpha k_1\varphi _1+\beta k_2 \varphi _2)-\frac{1}{2}(\alpha F_1(x)\varphi _1+\beta G_1(x)\varphi _2)\right\} dx. \end{aligned}$$

On the other hand, from $(H_{2}(i))$ we obtain

$$\begin{aligned} \frac{1}{p}\int _{\Omega }(\alpha f^{+\infty }\varphi _1+\beta g^{+\infty }\varphi _2)dx<\frac{1}{p}\int _{\Omega }\left( \alpha k_1\varphi _1+\beta k_2 \varphi _2\right) dx. \end{aligned}$$

It follows from $H_{2}(ii)$ that

$$\begin{aligned}&\int _{\Omega }\left\{ \frac{1}{2}(\alpha F_1(x)\varphi _1+\beta G_1(x)\varphi _2)-\frac{\alpha }{p} f^{+\infty }(x)\varphi _1-\frac{\beta }{p} g^{+\infty }(x)\varphi _2\right\} dx\\&\quad >\left( 1-\frac{1}{p}\right) \int _{\Omega }(\alpha k_1\varphi _1+\beta k_2 \varphi _2)dx. \end{aligned}$$

Thus,

$$\begin{aligned} \int _{\Omega }\left\{ \frac{1}{2}(\alpha F_1(x)\varphi _1+\beta G_1(x)\varphi _2)-(\alpha k_1\varphi _1+\beta k_2 \varphi _2)\right\} dx>0. \end{aligned}$$

This shows that

$$\begin{aligned} \underset{t\rightarrow +\infty }{\text {lim}}I(t\varphi )=-\infty . \end{aligned}$$

Next, with $t<0$ we also have

$$\begin{aligned} \frac{H{(x,t\varphi )}}{t}&= \frac{1}{t}\left\{ \frac{\alpha }{2}\int _0^{t\varphi _1}(f(x,s,t\varphi _2)+f(x,s,0))ds\right. \\&\left. +\frac{\beta }{2}\int _0^{t\varphi _2}(g(x,t\varphi _1,\tau )+g(x,0,\tau ))d\tau \right\} \\&= -\frac{1}{|t|}\left\{ \frac{\alpha }{2}\int _0^{-|t|\varphi _1}(f(x,s,-|t|\varphi _2)+f(x,s,0))ds\right. \\&\left. +\frac{\beta }{2}\int _0^{-|t|\varphi _2}(g(x,-|t|\varphi _1,\tau )+g(x,0,\tau ))d\tau \right\} . \end{aligned}$$

Set $s=-y\varphi _1 \rightarrow ds=-\varphi _1 dy$ and $s=-|t|\varphi _1=-y\varphi _1\Rightarrow y=|t|$

$$\begin{aligned} \frac{H{(x,t\varphi )}}{t}&= -\frac{1}{|t|}\left\{ \frac{\alpha }{2}\int _0^{-|t|}((f(x,-y\varphi _1,-|t|\varphi _2)+f(x,-y\varphi _1,0))dy)(-\varphi _1)\right. \\&\left. +\frac{\beta }{2}\int _0^{-|t|}((g(x,-|t|\varphi _1,-y\varphi _2)+g(x,0,-y\varphi _2))dy)(-\varphi _2)\right\} . \end{aligned}$$

Now, letting $t\rightarrow -\infty$, we get

$$\begin{aligned} \underset{t\rightarrow -\infty }{\text {lim}} \frac{H{(x,t\varphi )}}{t}=\frac{1}{2}\int _{\Omega }(\alpha F_2(x)\varphi _1+\beta G_2(x)\varphi _2)dx. \end{aligned}$$

We deduce that

$$\begin{aligned} \underset{t\rightarrow -\infty }{\text {lim}} I(t\varphi )=\underset{t\rightarrow -\infty }{\text {lim}}t\int _{\Omega }\left\{ (\alpha k_1 \varphi _1+\beta k_2 \varphi _2)-\frac{1}{2}(\alpha F_2(x)\varphi _1+\beta G_2(x)\varphi _2)\right\} dx. \end{aligned}$$

Similarly above from $(H_2(ii))$ we obtain

$$\begin{aligned} \frac{1}{2}\int _{\Omega }(\alpha F_2(x)\varphi _1+\beta G_2(x)\varphi _2)dx<\int _{\Omega }(\alpha k_1 \varphi _1+\beta k_2 \varphi _2)dx. \end{aligned}$$

This implies that

$$\begin{aligned} \underset{t\rightarrow -\infty }{\text {lim}} I(t\varphi )=-\infty . \end{aligned}$$

Thus, there exists $t_0$ such that $|t_0|$ large enough, we have $I(t_0\varphi )<0$.

Set $w_0(x)=(t_0\varphi _1,t_0\varphi _2)$ we get

$$\begin{aligned} I(w_0)= I(t_0\varphi )<B_Y\le I(t\varphi ). \end{aligned}$$

Proof of theorem 1.1

By Propositions 2.1 and 2.2, applying the Saddle Point Theorem (P.H.Rabinowitz) (see Theorem 2.1), we deduce that the functional $I$ attains its proper infimum at some $w_0=(u_0,v_0)\in E$, so that the problem (1.1) has at least a weak solution $w_0\in E$. Moreover $w_0$ is nontrivial weak solution of the Problem (1.1). The Theorem 1.1 is completely proved. $\square$

Remark 2.3

We will get the same result as above if the hypotheses $(H_2)$ is replaced by reverse inequalities as follows.

We assume that

$(H_2)^*$

$$\begin{aligned}&\int _{\Omega }\left\{ \frac{1}{2}(\alpha F_2(x)\varphi _1(x)+\beta G_2(x)\varphi _2(x))-\frac{\alpha }{p}f^{-\infty }(x)\varphi _1(x)-\frac{\beta }{p}g^{-\infty }(x)\varphi _2(x)\right\} dx\nonumber \\&\quad >\left( 1-\frac{1}{p}\right) \int _{\Omega }(\alpha k_1(x)\varphi _1(x)+\beta k_2(x)\varphi _2(x))dx>\nonumber \\&\quad >\int _{\Omega }\left\{ \frac{1}{2}(\alpha F_1(x)\varphi _1(x)+\beta G_1(x)\varphi _2(x))-\frac{\alpha }{p}f^{+\infty }(x)\varphi _1(x)-\frac{\beta }{p}g^{+\infty }(x)\varphi _2(x)\right\} dx.\nonumber \\ \end{aligned}$$

(2.27)

This means that, if the conditions $(H_1)$, $(H_2)^*$ holds, then the problem (1.1) has at least a nontrivial weak solution in $E$. This assertion is proved by using variational techniques, the Minimum Principle and generalization of the Landesman–Lazer type condition.