Singular quasilinear elliptic systems involving gradient terms

TL;DR

This paper proves the existence of smooth positive solutions for a class of singular quasilinear elliptic systems that include gradient terms, using a combination of sub-supersolutions and Schauder's fixed point theorem.

Contribution

It introduces a novel approach combining sub-supersolutions and Schauder's theorem to handle singular quasilinear elliptic systems with gradient terms.

Findings

Existence of smooth positive solutions established

Method applicable to singular systems with gradient dependence

Framework extends previous results in elliptic systems

Abstract

In this paper we establish existence of smooth positive solutions for a singular quasilinear elliptic system involving gradient terms. The approach combines sub-supersolutions method and Schauder's fixed point theorem.

Equations

$$P$$

$$\begin{array}[]{c}m_{1}s_{1}^{\alpha_{1}}s_{2}^{\beta_{1}}\leq f(x,s_{1},s_{2},\xi_{1},\xi_{2})\leq M_{1}s_{1}^{\alpha_{1}}s_{2}^{\beta_{1}}+|\xi_{1}|^{\gamma_{1}}+|\xi_{2}|^{\theta_{1}}\end{array}$$

$$0\le {\alpha }_1+{\beta }_1<-{\alpha }_1+{\beta }_1<p-1\text{ and }\max \{{\gamma }_1,{\theta }_1\}<p-1.$$

$$\begin{array}[]{c}m_{2}s_{1}^{\alpha_{2}}s_{2}^{\beta_{2}}\leq g(x,s_{1},s_{2},\xi_{1},\xi_{2})\leq M_{2}s_{1}^{\alpha_{2}}s_{2}^{\beta_{2}}+|\xi_{1}|^{\gamma_{2}}+|\xi_{2}|^{\theta_{2}}\end{array}$$

$$0\le {\alpha }_2+{\beta }_2<-{\alpha }_2+{\beta }_2<q-1\text{ and }\max \{{\gamma }_2,{\theta }_2\}<q-1.$$

$$\begin{array}[]{c}\tilde{c}_{0}d(x)\leq u(x)\leq\tilde{c}_{1}d(x)\text{ \ and \ }\tilde{c}_{0}^{\prime}d(x)\leq v(x)\leq\tilde{c}_{1}^{\prime}d(x)\text{ in }\Omega,\end{array}$$

$$l_1{\varphi }_{1,p}(x)\le {\varphi }_{1,q}(x)\le l_2{\varphi }_{1,p}(x)\text{ for all }x\in \Omega$$

$$\hat{l}d(x)\ge {\varphi }_{1,p}(x),{\varphi }_{1,q}(x)\ge ld(x)\text{ for all }x\in \Omega \text{.}$$

$$[u_1,u_2]\times [v_1,v_2]=\{(u,v)\in W_0^{1,p}(\Omega )\times W_0^{1,q}(\Omega ):u_1\le u\le u_2,v_1\le v\le v_2\text{a.e. in }\Omega \}.$$

$${\int }_{\Omega }|\nabla u{|}^{p-2}\nabla u\nabla \phi dx={\int }_{\Omega }f(x,u,v,\nabla u,\nabla v)\phi dx,$$

$${\int }_{\Omega }|\nabla v{|}^{q-2}\nabla v\nabla \psi dx={\int }_{\Omega }g(x,u,v,\nabla u,\nabla v)\psi dx$$

$$P_{(f_1,f_2)}$$

$$\left\{\begin{array}[]{l}\int_{\Omega}\left|\nabla\underline{u}\right|^{p-2}\nabla\underline{u}\nabla\varphi\ dx-\int_{\Omega}f_{1}(x,\underline{u},w_{2})\varphi\ dx\leq 0\\ \int_{\Omega}\left|\nabla\underline{v}\right|^{q-2}\nabla\underline{v}\nabla\psi\ dx-\int_{\Omega}f_{2}(x,w_{1},\underline{v})\psi\ dx\leq 0,\end{array}\right.$$

$$\left\{\begin{array}[]{l}\int_{\Omega}\left|\nabla\overline{u}\right|^{p-2}\nabla\overline{u}\nabla\varphi\ dx-\int_{\Omega}f_{1}(x,\overline{u},w_{2})\varphi\ dx\geq 0\\ \int_{\Omega}\left|\nabla\overline{v}\right|^{q-2}\nabla\overline{v}\nabla\psi\ dx-\int_{\Omega}f_{2}(x,w_{1},\overline{v})\psi\ dx\geq 0,\end{array}\right.$$

$$\underline{u}=\max \{{\underline{u}}_1,{\underline{u}}_2\},\underline{v}=\max \{{\underline{v}}_1,{\underline{v}}_2\},$$

$$\tilde{u}=\min \{{\overline{u}}_1,{\overline{u}}_2\},\tilde{v}=\min \{{\overline{v}}_1,{\overline{v}}_2\}$$

$$\underline{u}\le \tilde{u},\underline{v}\le \tilde{v}.$$

$$f_1(x,w_1,w_2)\in W^{-1,p^{\prime }}(\Omega ),f_2(x,w_1,w_2)\in W^{-1,q^{\prime }}(\Omega )$$

$$\nabla {\xi }_{\epsilon }(({\overline{u}}_1-{\overline{u}}_2{)}^{-})={\xi }_{\epsilon }^{\prime }(({\overline{u}}_1-{\overline{u}}_2{)}^{-})\nabla ({\overline{u}}_1-{\overline{u}}_2{)}^{-}$$

$$\nabla {\xi }_{\epsilon }(({\underline{u}}_1-{\underline{u}}_2{)}^{+})={\xi }_{\epsilon }^{\prime }(({\underline{u}}_1-{\underline{u}}_2{)}^{+})\nabla ({\underline{u}}_1-{\underline{u}}_2{)}^{+}.$$

$$⟨-{\Delta }_p{\overline{u}}_1,{\xi }_{\epsilon }(({\overline{u}}_1-{\overline{u}}_2{)}^{-})\phi ⟩\ge {\int }_{\Omega }{f}_1(x,{\overline{u}}_1,{\hat{w}}_2){\xi }_{\epsilon }(({\overline{u}}_1-{\overline{u}}_2{)}^{-})\phi dx,$$

$$⟨-{\Delta }_p{\underline{u}}_1,{\xi }_{\epsilon }(({\underline{u}}_1-{\underline{u}}_2{)}^{+})\phi ⟩\le {\int }_{\Omega }{f}_1(x,{\underline{u}}_1,{\hat{w}}_2){\xi }_{\epsilon }(({\underline{u}}_1-{\underline{u}}_2{)}^{+})\phi dx,$$

$$⟨-{\Delta }_p{\overline{u}}_2,(\epsilon -{\xi }_{\epsilon }(({\overline{u}}_1-{\overline{u}}_2{)}^{-}))\phi ⟩\ge {\int }_{\Omega }{f}_1(x,{\overline{u}}_2,{\check{w}}_2)\left(\epsilon -{\xi }_{\epsilon }(({\overline{u}}_1-{\overline{u}}_2{)}^{-})\right)\phi dx,$$

$$⟨-{\Delta }_p{\underline{u}}_2,(\epsilon -{\xi }_{\epsilon }(({\underline{u}}_1-{\underline{u}}_2{)}^{+}))\phi ⟩\le {\int }_{\Omega }{f}_1(x,{\underline{u}}_2,{\check{w}}_2)\left(\epsilon -{\xi }_{\epsilon }(({\underline{u}}_1-{\underline{u}}_2{)}^{+})\right)\phi dx,$$

$$\begin{array}[]{l}\left\langle-\Delta_{p}\overline{u}_{1},\xi_{\varepsilon}((\overline{u}_{1}-\overline{u}_{2})^{-})\varphi\right\rangle+\left\langle-\Delta_{p}\overline{u}_{2},(\varepsilon-\xi_{\varepsilon}((\overline{u}_{1}-\overline{u}_{2})^{-}))\varphi\right\rangle\\ \leq\int_{\Omega}|\nabla\overline{u}_{1}|^{p-2}(\nabla\overline{u}_{1},\nabla\varphi)_{\mathbb{R}^{N}}\xi_{\varepsilon}((\overline{u}_{1}-\overline{u}_{2})^{-})\text{ }dx\\ +\int_{\Omega}|\nabla\overline{u}_{2}|^{p-2}(\nabla\overline{u}_{2},\nabla\varphi)_{\mathbb{R}^{N}}\left(\varepsilon-\xi_{\varepsilon}((\overline{u}_{1}-\overline{u}_{2})^{-})\right)\text{ }dx\end{array}$$

$$\begin{array}[]{l}\left\langle-\Delta_{p}\underline{u}_{1},(\xi_{\varepsilon}((\underline{u}_{1}-\underline{u}_{2})^{+}))\varphi\right\rangle+\left\langle-\Delta_{p}\underline{u}_{2},\varepsilon-\xi_{\varepsilon}((\underline{u}_{1}-\underline{u}_{2})^{+})\varphi\right\rangle\\ \geq\int_{\Omega}|\nabla\underline{u}_{1}|^{p-2}(\nabla\underline{u}_{1},\nabla\varphi)_{\mathbb{R}^{N}}\xi_{\varepsilon}((\underline{u}_{1}-\underline{u}_{2})^{+})\text{ }dx\\ +\int_{\Omega}|\nabla\underline{u}_{2}|^{p-2}(\nabla\underline{u}_{2},\nabla\varphi)_{\mathbb{R}^{N}}\left(\varepsilon-\xi_{\varepsilon}((\underline{u}_{1}-\underline{u}_{2})^{+})\right)\text{ }dx.\end{array}$$

$$\begin{array}[]{l}\int_{\Omega}|\nabla\overline{u}_{1}|^{p-2}(\nabla\overline{u}_{1},\nabla\varphi)_{\mathbb{R}^{N}}\frac{1}{\varepsilon}\xi_{\varepsilon}((\overline{u}_{1}-\overline{u}_{2})^{-})\text{ }dx\\ +\int_{\Omega}|\nabla\overline{u}_{2}|^{p-2}(\nabla\overline{u}_{2},\nabla\varphi)_{\mathbb{R}^{N}}\left(1-\frac{1}{\varepsilon}\xi_{\varepsilon}((\overline{u}_{1}-\overline{u}_{2})^{-})\right)\text{ }dx\\ \geq\int_{\Omega}f_{1}(x,\overline{u}_{1},w_{2})\frac{1}{\varepsilon}\xi_{\varepsilon}((\overline{u}_{1}-\overline{u}_{2})^{-})\varphi\text{ }dx+\int_{\Omega}f_{1}(x,\overline{u}_{2},w_{2})\left(1-\frac{1}{\varepsilon}\xi_{\varepsilon}((\overline{u}_{1}-\overline{u}_{2})^{-})\right)\varphi\text{ }dx,\end{array}$$

$$\begin{array}[]{l}\int_{\Omega}|\nabla\underline{u}_{1}|^{p-2}(\nabla\underline{u}_{1},\nabla\varphi)_{\mathbb{R}^{N}}\frac{1}{\varepsilon}\xi_{\varepsilon}((\underline{u}_{1}-\underline{u}_{2})^{+})\text{ }dx\\ +\int_{\Omega}|\nabla\underline{u}_{2}|^{p-2}(\nabla\underline{u}_{2},\nabla\varphi)_{\mathbb{R}^{N}}\left(1-\frac{1}{\varepsilon}\xi_{\varepsilon}((\underline{u}_{1}-\underline{u}_{2})^{+})\right)\text{ }dx\\ \leq\int_{\Omega}f_{1}(x,\underline{u}_{1},w_{2})\frac{1}{\varepsilon}\xi_{\varepsilon}((\underline{u}_{1}-\underline{u}_{2})^{+})\varphi\text{ }dx+\int_{\Omega}f_{1}(x,\underline{u}_{2},w_{2})\left(1-\frac{1}{\varepsilon}\xi_{\varepsilon}((\underline{u}_{1}-\underline{u}_{2})^{+})\right)\varphi\text{ }dx,\end{array}$$

$$\left\{\begin{array}[]{c}\frac{1}{\varepsilon}\xi_{\varepsilon}((\overline{u}_{1}-\overline{u}_{2})^{-}\rightarrow\chi_{\{\overline{u}_{1}<\overline{u}_{2}\}}(x)\\ \frac{1}{\varepsilon}\xi_{\varepsilon}((\underline{u}_{1}-\underline{u}_{2})^{+}\rightarrow\chi_{\{\underline{u}_{1}>\underline{u}_{2}\}}(x)\end{array}\right.\text{, \ a.e. in }\Omega\text{ as }\varepsilon\rightarrow 0,$$

$${\int }_{\Omega }|\nabla \tilde{u}{|}^{p-2}\nabla \tilde{u}\nabla \phi dx\ge {\int }_{\Omega }{f}_1(x,\tilde{u},w_2)\phi dx$$

Full text

1. Introduction

Let $\Omega\subset\mathbb{R}^{N}$ $\left(N\geq 2\right)$ be a bounded domain with smooth boundary $\partial\Omega$. We deal with the following quasilinear elliptic system

[TABLE]

where $\Delta_{p}$ (resp. $\Delta_{q})$ stands for the $p$-Laplacian (resp. $q$-Laplacian) differential operator on $W_{0}^{1,p}(\Omega)$ (resp. $W_{0}^{1,q}(\Omega)$) with $1<p,q\leq N$. The nonlinearity terms $f(x,u,v,\nabla u,\nabla v)$ and $g(x,u,v,\nabla u,\nabla v),$ which is often expressed as dealing with convection terms, can exhibit singularities when the variables $u$ and $v$ approach zero. Specifically, we assume that $f,g:\Omega\times(0,+\infty)\times(0,+\infty)\times\mathbb{R}^{2N}\rightarrow(0,+\infty)$ are Carathéodory functions, that is, $f(\cdot,s_{1},s_{2},\xi_{1},\xi_{2})$ and $g(\cdot,s_{1},s_{2},\xi_{1},\xi_{2})$ are measurable for every $(s_{1},s_{2},\xi_{1},\xi_{2})\in(0,+\infty)\times(0,+\infty)\times\mathbb{R}^{2N}$ and $f(x,\cdot,\cdot,\cdot,\cdot)$ and $g(x,\cdot,\cdot,\cdot,\cdot)$ are continuous functions for a.e. $x\in\Omega,$ and are subjected to the hypotheses:

$\mathrm{H}(f)$**: **

There exist constants $M_{1},m_{1}>0$ and $-1<\alpha_{1}<0<\beta_{1},\gamma_{1},\theta_{1}$ such that

[TABLE]

for a.e. $x\in\Omega$, for all $s_{1},s_{2}>0$, for all $\xi_{1},\xi_{2}\in\mathbb{R}^{N}$, with

[TABLE]

$\mathrm{H}(g)$**: **

There exist constants $M_{2},m_{2}>0$ and $-1<\beta_{2}<0<\alpha_{2},\gamma_{2},\theta_{2}$ such that

[TABLE]

for a.e. $x\in\Omega$, for all $s_{1},s_{2}>0$, for all $\xi_{1},\xi_{2}\in\mathbb{R}^{N}$, with

[TABLE]

The main interest of this work lies in the dependence of the right hand side terms on the solution and its gradient. The presence of the latter constitutes a serious obstacle in the study of the problem ($P$). Namely, the imposed hypotheses do not guarantee that the structure of the system is variational. Thus, variational methods cannot be applied. Another important aspect of problem ($P$) is that the convection terms can exhibit singularities when the variables $u$ and $v$ approach zero. This occur under hypotheses $\mathrm{H}(f)$ and $\mathrm{H}(g)$ where exponents $\alpha_{1}$ and $\beta_{2}$ are allowed to be negative. This type of problem is rare in the literature. Actually, according to our knowledge, singular system ($P$) was examined only in [24] where the system is supposed to have a competitive structure. This means that the nonlinearities $f$ and $g$ are not increasing with respect to $v$ and $u$, respectively. Beside that, the singularities appear in both the solution and its gradient through some specific growth conditions. These combined with properties of the eigenfunction corresponding to the first eigenvalue of the operators $-\Delta_{p}$ and $-\Delta_{q}$ is a key point on which the existence results is proved. It is worth pointing out that the assumptions imposed therein, precisely (1.2)-(1.5), are not satisfied for system ($P$) under hypotheses $\mathrm{H}(f)$ and $\mathrm{H}(g)$. Moreover, in this work, neither competitive nor the complementary situation called cooperative structure on the system ($P$) is imposed.

The semilinear case (i.e., $p=q=2$) for a class of singular systems with convection terms was examined by Alves, Carriao and Faria [3], and by Alves and Moussaoui [4], by essentially using the linearity of the principal part. For singular elliptic systems without gradient terms, we refer to Alves and Corrêa [1], Alves, Corrêa and Gonçalves [2], El Manouni, Perera and Shivaji [9], Ghergu [11, 12], Hernández, Mancebo and Vega [15], Montenegro and Suarez [19], Motreanu and Moussaoui [21, 22, 23].

The main result of the present paper provides the existence of (positive) smooth solutions for the singular system ($P$).

Theorem 1**.**

Assume $\mathrm{H}(f)$ and $\mathrm{H}(g)$ hold. Then problem ($P$) admits a (positive) solution $(u,v)$ in $C_{0}^{1}(\overline{\Omega})\times C_{0}^{1}(\overline{\Omega})$ satisfying

[TABLE]

for some positive constants $\tilde{c}_{0},\tilde{c}_{0}^{\prime},\tilde{c}_{1}$ and $\tilde{c}_{1}^{\prime}$.

The proof of Theorem 1 is chiefly based on sub-supersolution method together with Schauder’s fixed point Theorem. However, the sub-supersolution method cannot be directly implemented. On the one hand, this is due to the presence of singular terms in system ($P$). In this respect, it should be pointed out that, to the best of our knowledge, there is no theorem involving gradient terms and singularities which garantees the existence of a solution within a sub-supersolution pair. On the other hand, the dependence of the right hand side terms on the gradient of the solution complicates further the application of the above method to the problem ($P$). Specifically, the definition of sub-supersolutions pairs for system ($P$) (see [7]) seems to be hardly applicable because, a priori, no conclusion can be drawn on the comparison of the gradient of two comparable functions. To handle problem ($P$), we consider an auxiliary system for which, the sub-supersolution Theorem involving singular terms in [16, Theorem 2.1] is applicable. Here, we construct the sub and supersolution pair by choosing suitable functions with an adjustment of adequate constants. Then, focusing on the rectangle formed by these functions, we prove the existence of a smallest and a biggest positive solutions of the auxiliary problem. The argument is based on the Hardy-Sobolev inequality, Zorn’s Lemma and the the $S_{+}$-property of the negative $p$-Laplacian operator on $W_{0}^{1,p}(\Omega)$. Thereby, these allow to construct a suitable operator whose fixed points, obtained via Schauder’s fixed point theorem, are exactly solutions of ($P$).

The rest of the paper is organized as follows. Section 2 presents auxiliary results related to sub-supersolutions and extremal solutions. Section 3 deals with the existence of a smallest positive solution for an auxialiary system. Section 4 contains the proof of the main result.

2. Preliminary results

Given $1<p<+\infty$, the spaces $L^{p}(\Omega)$ and $W_{0}^{1,p}(\Omega)$ are endowed with the usual norms $\|u\|_{p}=(\int_{\Omega}|u|^{p}\ dx)^{1/p}$ and $\|u\|_{1,p}=(\int_{\Omega}|\nabla u|^{p}\ dx)^{1/p}$, respectively. Denote by $p^{\prime}=\frac{p}{p-1}$ and $q^{\prime}=\frac{q}{q-1}$. We will also utilize the spaces $C(\overline{\Omega})$ and C_{0}^{1,\beta}(\overline{\Omega})=\{u\in C^{1,\beta}(\overline{\Omega}):u=0\ \mbox{on \partial\Omega}\} with $\beta\in(0,1)$.

For later use, we denote by $\lambda_{1,p}$ and $\lambda_{1,q}$ the first eigenvalue of $-\Delta_{p}$ on $W_{0}^{1,p}(\Omega)$ and of $-\Delta_{q}$ on $W_{0}^{1,q}(\Omega)$, respectively.

Let $\phi_{1,p}$ be the positive eigenfunction of $-\Delta_{p}$ corresponding to $\lambda_{1,p}$, that is $-\Delta_{p}\phi_{1,p}=\lambda_{1,p}\phi_{1,p}^{p-1}$  in $\Omega,$  $\phi_{1,p}=0$  on $\partial\Omega$. Similarly, let $\phi_{1,q}$ be the positive eigenfunction of $-\Delta_{q}$ corresponding to $\lambda_{1,q}$, that is $-\Delta_{q}\phi_{1,q}=\lambda_{1,q}\phi_{1,q}^{q-1}$  in $\Omega,$ $\phi_{1,q}=0$  on $\partial\Omega$. The strong maximum principle ensures the existence of positive constants $l_{1}$, $l_{2}$, $\hat{l}$ and $l$ such that (see also [13])

[TABLE]

and

[TABLE]

Here, $d(x)$ denotes the distance from a point $x\in\overline{\Omega}$ to the boundary $\partial\Omega$, where $\overline{\Omega}=\Omega\cup\partial\Omega$ is the closure of $\Omega\subset\mathbb{R}^{N}$.

Throughout the paper, if $(u_{1},v_{1})$, $(u_{2},v_{2})\in W_{0}^{1,p}(\Omega)\times W_{0}^{1,q}(\Omega)$ are such that $u_{1}\leq u_{2}$ and $v_{1}\leq v_{2}$ a.e. in $\Omega$ we will write $(u_{1},v_{1})\leq(u_{2},v_{2})$ and we will use the notation

[TABLE]

A (weak) solution of ($P$) is any pair $(u,v)\in W_{0}^{1,p}(\Omega)\times W_{0}^{1,q}(\Omega)$ such that

[TABLE]

[TABLE]

for all $(\varphi,\psi)\in W_{0}^{1,p}(\Omega)\times W_{0}^{1,q}(\Omega)$.

We will study auxiliary problems with not convection terms, for this reason let us consider the following quasilinear elliptic problem

[TABLE]

where $f_{i}:\Omega\times(0,+\infty)\times(0,+\infty)\rightarrow\mathbb{R}$, $i=1,2$, are Carathéodory functions which can exhibit singularities near zero.

Recall that $\left(\underline{u},\underline{v}\right)$, $\left(\overline{u},\overline{v}\right)\in(W^{1,p}(\Omega)\cap L^{\infty}(\Omega))\times(W^{1,q}(\Omega)\cap L^{\infty}(\Omega))$ form a pair of a sub-supersolution for ($P_{(f_{1},f_{2})}$) if $(\underline{u},\underline{v})\leq(\overline{u},\overline{v})$ and

[TABLE]

[TABLE]

for all $(\varphi,\psi)\in W_{0}^{1,p}\left(\Omega\right)\times W_{0}^{1,q}\left(\Omega\right)$ with $\varphi,\psi\geq 0$ a.e. in $\Omega$ and for all $\left(w_{1},w_{2}\right)\in[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$.

Lemma 1**.**

Let $(\underline{u}_{i},\underline{v}_{i}),(\overline{u}_{i},\overline{v}_{i})\in(W^{1,p}(\Omega)\cap L^{\infty}(\Omega))\times(W^{1,q}(\Omega)\cap L^{\infty}(\Omega)),$ for $i=1,2$. Put

[TABLE]

[TABLE]

and assume that

[TABLE]

Moreover, suppose that

[TABLE]

*for every $(w_{1},w_{2})\in[\underline{u},\tilde{u}]\times[\underline{v},\tilde{v}]$ and $(\underline{u}_{i},\underline{v}_{i})$, $(\overline{u}_{i},\overline{v}_{i})$ ($i=1,2$) form two pairs of sub-supersolutions for problem ($P_{(f_{1},f_{2})}$).

Then $(\underline{u},\underline{v})$, $(\tilde{u},\tilde{v})\in(W^{1,p}(\Omega)\cap L^{\infty}(\Omega))\times(W^{1,q}(\Omega)\cap L^{\infty}(\Omega))$ form also a pair of sub-supersolution for the problem ($P_{(f_{1},f_{2})}$).*

Proof.

Inspired by the proof of [20, Lemma 3], for a fixed $\varepsilon>0,$ let us define the truncation function $\xi_{\varepsilon}(s)=\max\{-\varepsilon,\min\{s,\varepsilon\}\}$ for $s\in\mathbb{R}.$ It is shown in [18] that $\xi_{\varepsilon}((\overline{u}_{1}-\overline{u}_{2})^{-}),\xi_{\varepsilon}((\underline{u}_{1}-\underline{u}_{2})^{+})\in W^{1,p}(\Omega),$

[TABLE]

and

[TABLE]

For any test function $\varphi\in C_{c}^{1}(\Omega)$ with $\varphi\geq 0,$ it holds

[TABLE]

[TABLE]

for all $\hat{w}_{2}\in W^{1,q}(\Omega)$ with $\underline{v}_{1}\leq\hat{w}_{2}\leq\overline{v}_{1}$, and

[TABLE]

[TABLE]

for all $\check{w}_{2}\in W^{1,q}(\Omega)$ with $\underline{v}_{2}\leq\check{w}_{2}\leq\overline{v}_{2}$. On the other hand, using the monotonicity of the $p$-Laplacian operator, we get

[TABLE]

and

[TABLE]

Then, gathering (2.3) together with (2.5) and (2.4) together with (2.6), by means of (2.7) and (2.8), one gets

[TABLE]

and

[TABLE]

for all $w_{2}\in W^{1,q}(\Omega)$ such that $\underline{v}\leq w_{2}\leq\tilde{v}$ a.e. in $\Omega$. Passing to the limit as $\varepsilon\rightarrow 0$ and noticing that

[TABLE]

where $\chi_{\mathcal{A}}$ is the characteristic function of the set $\mathcal{A},$ we obtain

[TABLE]

and

[TABLE]

for all $\varphi\in C_{c}^{1}(\Omega),$ $\varphi\geq 0$ a.e. in $\Omega$ and for all $w_{2}\in W^{1,q}(\Omega)$ within $[\underline{v},\tilde{v}]$ a.e. in $\Omega$.

In the same manner we get

[TABLE]

and

[TABLE]

for all $\psi\in C_{c}^{1}(\Omega),$ $\psi\geq 0$ a.e. in $\Omega$ and for all $w_{1}\in W^{1,p}(\Omega)$ within $[\underline{u},\tilde{u}]$ a.e. in $\Omega$. Finally, since $C_{c}^{1}(\Omega)$ is dense in both $W^{1,p}(\Omega)$ and $W^{1,q}(\Omega)$, we achieve the desired conclusion. ∎

Theorem 2**.**

Let $\left(\underline{u},\underline{v}\right),$ $\left(\overline{u},\overline{v}\right)\in C^{1}(\overline{\Omega})\times C^{1}(\overline{\Omega})$ be a pair of sub-supersolution ($P_{(f_{1},f_{2})}$) with $\underline{u},\underline{v}\geq c_{0}d(x)$ in $\Omega$ for some constant $c_{0}>0$ and suppose there exist constants $k_{1},k_{2}>0$ and $-1<\alpha,\beta<0$ such that

[TABLE]

*a.e. in $\Omega$ and for every $(u,v)\in[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$.

Then problem ($P_{(f_{1},f_{2})}$) has a smallest solution $(u^{\ast},v^{\ast})$ and a biggest solution $(u^{+},v^{+})$ in $C_{0}^{1,\gamma}(\overline{\Omega})\times C_{0}^{1,\gamma}(\overline{\Omega})$ for certain $\gamma\in(0,1)$, within $[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$.*

Proof.

We only prove the existence of a smallest positive solution $(u^{\ast},v^{\ast})\in C_{0}^{1,\gamma}(\overline{\Omega})\times C_{0}^{1,\gamma}(\overline{\Omega})$ within $[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$. That of a biggest positive solution within $[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$ can be carried out in the similar way.

Denote by $S$ the set of all $(w_{1},w_{2})\in[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$ that are solutions of ($P_{(f_{1},f_{2})}$). It is well know from [16, Theorem 2.1] that under assumption (2.9), system ($P_{(f_{1},f_{2})}$) has a (positive) solution $(u,v)\in C_{0}^{1,\gamma}(\overline{\Omega})\times C_{0}^{1,\gamma}(\overline{\Omega})$ for certain $\gamma\in(0,1)$, located in $[\underline{u},\overline{u}]\times[\underline{v},\overline{v}].$ Thus, $S$ is not empty. Moreover, let $(u_{1},v_{1}),(u_{2},v_{2})\in S$. Since $(\underline{u},\underline{v})$, $(u_{1},v_{1})$ and $(\underline{u},\underline{v})$, $(u_{2},v_{2})$ form two pairs of sub-supersolution, if we put $(\tilde{u},\tilde{v})=(\min\{u_{1},u_{2}\},\min\{v_{1},v_{2}\})\in(W_{0}^{1,p}(\Omega)\cap L^{\infty}(\Omega))\times(W_{0}^{1,q}(\Omega)\cap L^{\infty}(\Omega))$, by virtue of Lemma 1, $(\underline{u},\underline{v})$, $(\tilde{u},\tilde{v})$ form a pair of sub-supersolution for ($P_{(f_{1},f_{2})}$). Then, owing to [16, Theorem 2.1], there exists a solution of ($P_{(f_{1},f_{2})}$) in $([\underline{u},\tilde{u}]\cap C_{0}^{1}(\overline{\Omega}))\times([\underline{v},\tilde{v}]\cap C_{0}^{1}(\overline{\Omega}))$, which proves that $S$ is downward directed.

Now, let us consider a chain $C$ in $S$. Then there is a sequence $\{(u_{k},v_{k})\}_{k\geq 1}\subset C$ such that $\inf C=\inf_{k\geq 1}(u_{k},v_{k})$ (see [8, pag. 336]) and it is not restrictive assume $\{(u_{k},v_{k})\}_{k\geq 1}$ to be decreasing. Hence, if we put $(\hat{u},\hat{v})=\inf C$, one has that $u_{k}\to\hat{u}$ and $v_{k}\to\hat{v}$ a.e. in $\Omega$, that is

[TABLE]

Moreover, because $(u_{k},v_{k})$ for $k\geq 1$ are solutions of ($P_{(f_{1},f_{2})}$) we have

[TABLE]

and

[TABLE]

Since $-1<\alpha,\beta<0$, by virtue of the Hardy-Sobolev inequality (see, e.g., [1] or [25]), the last integrals in (2.11) and (2.12) are finite which in turn imply that $\{u_{k}\}$ and $\{v_{k}\}$ are bounded in $W_{0}^{1,p}(\Omega)$ and $W_{0}^{1,q}(\Omega),$ respectively. So, passing to relabelled subsequences and recalling the Rellich embedding theorem, we have

[TABLE]

Using $\varphi=u_{k}-\hat{u}$ and $\psi=v_{k}-\hat{v}$ as test functions we find that

[TABLE]

and

[TABLE]

From (2.9) and since $(u_{k},v_{k})\in S$ for all $k\in\mathbb{N}$, in view of (2.10) we have

[TABLE]

and

[TABLE]

Thank’s to [17, Lemma], we deduce that $f_{1}(x,u_{k},v_{k})(u_{k}-\hat{u})$ and $f_{2}(x,u_{k},v_{k})(v_{k}-\hat{v})$ are dominated by $L^{1}(\Omega)$ functions and using the dominated convergence theorem, we obtain

[TABLE]

Then the $S_{+}$-property of $-\Delta_{p}$ and $-\Delta_{q}$ on $W_{0}^{1,p}(\Omega)$ and $W_{0}^{1,q}(\Omega),$ respectively, guarantees that

[TABLE]

and therefore $(\hat{u},\hat{v})$ is a positive solution of problem ($P_{(f_{1},f_{2})}$). Consequently, $(\hat{u},\hat{v})=\inf C$ belongs to $S$. Then Zorn’s Lemma can be applied which provides a minimal element $(u^{*},v^{*})$ of $S$. Furthermore, since $S\subset[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$, (2.9) enables us to apply the regularity theory (see [14]) to infer that $(u^{*},v^{*})\in C_{0}^{1,\gamma}(\overline{\Omega})\times C_{0}^{1,\gamma}(\overline{\Omega})$ for some $\gamma\in(0,1)$.

The proof is completed by showing that $(u^{\ast},v^{\ast})$ is the smallest solution of ($P_{(f_{1},f_{2})}$) in $S$. To this end, let $(u,v)\in S$. Bearing in mind that $S$ is downward directed, there is $(\mathring{u},\mathring{v})\in S$ with $\mathring{u}\leq u^{\ast},$ $\mathring{v}\leq v^{\ast}$ and $\mathring{u}\leq u,$ $\mathring{v}\leq v$. Since $(u^{\ast},v^{\ast})$ is a minimal element of $S$, it turns out that $(u^{\ast},v^{\ast})=(\mathring{u},\mathring{v})\leq(u,v)$. The same reasoning can be used to prove the existence of a biggest solution $(u^{+},v^{+})$ in $C_{0}^{1,\gamma}(\overline{\Omega})\times C_{0}^{1,\gamma}(\overline{\Omega}),$ for certain $\gamma\in(0,1)$, within $[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$. This completes the proof. ∎

3. Auxiliary system

For every $z_{1},z_{2}\in C_{0}^{1}(\Omega),$ let us state the auxiliary problem

[TABLE]

With the aim of finding pairs of sub-supersolutions of problem ($P_{(z_{1},z_{2})}$), let us define $\xi_{1}$ and $\xi_{2}$ in $C_{0}^{1,\beta}(\overline{\Omega}),$ $\beta\in(0,1)$, as the unique solutions of the problems

[TABLE]

respectively, which are known to satisfy

[TABLE]

with constants $c_{i},c_{i}^{\prime}>0$ (see [6]).

Set

[TABLE]

where $C>1$ is a constant that will be fixed large enough and denote by

[TABLE]

Obviously, as a consequence of the maximum principle, we have

[TABLE]

Recall from [5, Lemma 1] that if $h_{1},h_{2}\in L^{\infty}(\Omega)$ and $u\in W_{0}^{1,p}(\Omega),$ $v\in W_{0}^{1,q}(\Omega)$ are the weak solutions of problems

[TABLE]

there exist positive constants $K_{p}=K_{p}(p,N,\Omega)$ and $K_{q}=K_{q}(q,N,\Omega)$ such that

[TABLE]

Denote by

[TABLE]

Using the functions in (3.3), we introduce the sets

[TABLE]

and

[TABLE]

which are closed, bounded and convex in $C_{0}^{1}(\overline{\Omega})$.

Proposition 1**.**

Assume $\mathrm{H}(f)$ and $\mathrm{H}(g)$. Then, for $C>1$ sufficiently large and for every $(z_{1},z_{2})\in\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C)$, problem ($P_{(z_{1},z_{2})}$) has a smallest solution $(u^{\ast},v^{\ast})_{(z_{1},z_{2})}$ in $C_{0}^{1,\gamma}(\overline{\Omega})\times C_{0}^{1,\gamma}(\overline{\Omega})$, for certain $\gamma\in(0,1)$, within $[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$.

Proof.

The proof is related to Theorem 2. First, let us prove that

• Claim:

For every $(z_{1},z_{2})\in\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C)$, $(\underline{u},\underline{v})$, $(\bar{u},\bar{v})$ form a pair of sub-supersolution for ($P_{(z_{1},z_{2})}$) provided that $C$ is large enough.

Using $\mathrm{H}(f)$, $\mathrm{H}(g)$, (3.3), (3.4) and (2.1) we get

[TABLE]

and

[TABLE]

provided that $C>1$ is large enough (such that (3.5) holds too). Then, it is readily seen from $\mathrm{H}(f)$ and $\mathrm{H}(g)$ that

[TABLE]

and

[TABLE]

for all $\left(\varphi,\psi\right)\in W_{0}^{1,p}\left(\Omega\right)\times W_{0}^{1,q}\left(\Omega\right)$ with $\varphi,\psi\geq 0$ in $\Omega,$ for all $\left(w_{1},w_{2}\right)\in W^{1,p}(\Omega)\times W^{1,q}(\Omega)$ satisfying $\underline{u}\leq w_{1}\leq\overline{u}$ and $\underline{v}\leq w_{2}\leq\overline{v}$ a.e. in $\Omega$, and for $(z_{1},z_{2})\in\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C)$.

Now, taking into account (3.1), (3.2), (3.4), $\mathrm{H}(f)$ and $\mathrm{H}(g)$, we derive the estimates

[TABLE]

and

[TABLE]

provided that $C>1$ is sufficiently large. Consequently, it turns out that

[TABLE]

and

[TABLE]

for all $\left(\varphi,\psi\right)\in W_{0}^{1,p}\left(\Omega\right)\times W_{0}^{1,q}\left(\Omega\right)$ with $\varphi,\psi\geq 0$ in $\Omega$, for all $\left(w_{1},w_{2}\right)\in W^{1,p}(\Omega)\times W^{1,q}(\Omega)$ satisfying $\underline{u}\leq w_{1}\leq\overline{u}$ and $\underline{v}\leq w_{2}\leq\overline{v}$ a.e. in $\Omega$, and for $(z_{1},z_{2})\in\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C)$.

Putting together (3), (3), (3) and (3) we get the Claim.

Furthermore, for every $(z_{1},z_{2})\in\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C)$ and for every $(u,v)\in[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$, from $\mathrm{H}(f)$, $\mathrm{H}(g)$, (3.3), (3.4) and (2.2) we have the estimates

[TABLE]

and

[TABLE]

for some positive constants $C_{1}$ and $C_{2}$ independent from $u$, $v$, $z_{1}$ and $z_{2}$. Then, owing to Theorem 2, it follows that if $C>1$ is large enough (according to the Claim), for every $(z_{1},z_{2})\in\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C)$ problem ($P_{(z_{1},z_{2})}$) has a smallest solution $(u^{\ast},v^{\ast})_{(z_{1},z_{2})}\in C_{0}^{1,\gamma}(\overline{\Omega})\times C_{0}^{1,\gamma}(\overline{\Omega})$ for certain $\gamma\in(0,1)$, within $[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$. This complete the proof. ∎

Remark 1**.**

We wish explicitly to point out that from the proof of Proposition 1 one can derive an estimate of the largeness of $C>1$. In particular, the choice of $C$, that first of all is related to (3.5), is crucial for verifying the Claim and, as a consequence, that for every $(z_{1},z_{2})\in\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C)$ the set of the solutions of problem ($P_{(z_{1},z_{2})}$) is nonempty.

In what follows, $C>1$ will be assumed large enough such that for any $(z_{1},z_{2})\in\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C)$, put

[TABLE]

then $S_{(z_{1},z_{2})}\neq\emptyset$.

Lemma 2**.**

Assume $\mathrm{H}(f)$ and $\mathrm{H}(g)$ hold and let $C>1$ be large enough. If $\{(z_{1,n},z_{2,n})\}$ is a sequence in $\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C)$ such that $(z_{1,n},z_{2,n})\rightarrow(z_{1},z_{2})$ in $\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C)$, then for any $(\breve{u},\breve{v})\in S_{(z_{1},z_{2})}$, there exists $(\breve{u}_{n},\breve{v}_{n})\in S_{(z_{1,n},z_{2,n})}$ such that $(\breve{u}_{n},\breve{v}_{n})\rightarrow(\breve{u},\breve{v})$ in $C_{0}^{1}(\overline{\Omega})\times C_{0}^{1}(\overline{\Omega})$.

Proof.

Fix $C>1$ large enough, put

[TABLE]

and observe that

[TABLE]

uniformly in $x\in\Omega$, for all $t\geq\min\{\underline{u}(x),\underline{v}(x)\}$, where $\mu_{1}\in\{m_{1},\,M_{1}\}$ and $\mu_{2}\in\{m_{2},\,M_{2}\}$.

Indeed, from (2.2) and bearing in mind that $\alpha_{1}-p+1<0<\beta_{1}$ one has that

[TABLE]

for all $x\in\Omega$. Hence,

[TABLE]

for all $t\geq\min\{\underline{u}(x),\underline{v}(x)\}$ and uniformly in $\Omega$, so that the first inequality in (3.13) holds. The second inequality can be verified arguing in analogy.

Here, condition (3.13) guaranties that for all $(u,v)\in\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C)$ the functions

[TABLE]

are monotone with respect to $t\geq\min\{\underline{u}(x),\underline{v}(x)\}$.

Let now $\hat{f}$, $\hat{g}$ be the functions defined by

[TABLE]

[TABLE]

for $(x,s_{1},s_{2},\xi_{1},\xi_{2})\in\Omega\times(0,+\infty)\times(0,+\infty)\times\mathbb{R}^{2N}$.

Arguing as in (3.11) and (3.12), bearing in mind (3.2) and (3.3), there exist two positive constants $\hat{C}_{1}$ and $\hat{C}_{2}$ such that

[TABLE]

a.e. in $\Omega$, for every $(u,v)\in[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$ and every $(y_{1},y_{2})\in\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C)$.

Let us consider now the following differential operators $L_{p}:W_{0}^{1,p}(\Omega)\rightarrow W^{-1,p^{\prime}}(\Omega)$, $L_{q}:W_{0}^{1,q}(\Omega)\rightarrow W^{-1,q^{\prime}}(\Omega)$ defined by

[TABLE]

for all $u\in W_{0}^{1,p}(\Omega)$, and

[TABLE]

for all $u\in W_{0}^{1,q}(\Omega)$. Observing that $p^{\prime}(\alpha_{1}+\beta_{1})-p>-p$ and $q^{\prime}(\alpha_{2}+\beta_{2})-q>-q$, one can apply [25, Theorem 19.8] ($\partial\Omega$ is assumed to be smooth enough) in order to obtain

[TABLE]

namely $L_{p}$ and $L_{q}$ are well defined.

A direct computation shows that $L_{p}$ and $L_{q}$ are demicontinuous, coercive and strictly monotone. Hence, in view of (3.15), one can apply the Minty-Browder theorem and conclude that for every $(u,v)\in[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$, for every $(y_{1},y_{2})\in\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C)$ the problem

[TABLE]

admits a unique solution.

At this point fix $(z_{1},z_{2})\in\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C)$, $(\breve{u},\breve{v})\in S_{(z_{1},z_{2})}$ and let $\{(z_{1,n},z_{2,n})\}$ be a sequence in $\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C)$ such that $(z_{1,n},z_{2,n})\to(z_{1},z_{2})$.

Obviously, $(\breve{u},\breve{v})\in S_{(z_{1},z_{2})}$ implies that

[TABLE]

Fix $n\in\mathbb{N}$ and let $(w_{1,n}^{0},w_{2,n}^{0})$ be the unique solution of the problem $(P_{(\breve{u},\breve{v},z_{1,n},z_{2,n})})$.

By $\mathrm{H}(f)$ and $\mathrm{H}(g)$, since $(\breve{u},\breve{v})\in[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$, using the monotonicity of the functions introduced in (3.14) and the computations pointed out in (3) and (3), it follows that

[TABLE]

and similarly, we obtain

[TABLE]

The same reasoning can be exploited for assuring that

[TABLE]

Accordingly, the weak comparison principle in [26] implies that $(w_{1,n}^{0},w_{2,n}^{0})\in[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$. Furthermore, from (3.15), by the regularity theory (see [14, Lemma 3.1]), it follows $(w_{1,n}^{0},w_{2,n}^{0})\in C_{0}^{1,\gamma}(\overline{\Omega})\times C_{0}^{1,\gamma}(\overline{\Omega})$ for some $\gamma\in(0,1)$, and, in particular, $\{(w_{1,n},w_{2,n})\}$ is bounded in $C_{0}^{1,\gamma}(\overline{\Omega})\times C_{0}^{1,\gamma}(\overline{\Omega})$. Then, since $C_{0}^{1,\gamma}(\overline{\Omega})\subset C_{0}^{1}(\overline{\Omega})$ is compact, there exist a subsequence, denoted by the same symbol, $\{(w_{1,n}^{0},w_{2,n}^{0})\}$ and $(\hat{u},\hat{v})$ such that

[TABLE]

Hence, passing to the limit in $(P_{(\breve{u},\breve{v},z_{1,n},z_{2,n})})$), one has that $(\hat{u},\hat{v})$ is a solution of the problem $(P_{\breve{u},\breve{v},z_{1},z_{2}})$. Namely, in view of (3.16), $(\hat{u},\hat{v})=(\breve{u},\breve{v})$ and by the strong convergence (3.17) we infer that

[TABLE]

Now, let $(w_{1,n}^{1},w_{2,n}^{1})$ be the unique solution of the problem $(P_{w_{1,n}^{0},w_{2,n}^{0},z_{1,n},z_{2,n}})$. Following the same argument as before we obtain

[TABLE]

and

[TABLE]

Inductively, for each $n\in\mathbb{N}$, we construct the sequences $\{(w_{1,n}^{k},w_{2,n}^{k})\}_{k}$ in $C_{0}^{1}(\overline{\Omega})\times C_{0}^{1}(\overline{\Omega})$ as a unique solution of

[TABLE]

such that for each $k\in\mathbb{N},$ we have

[TABLE]

and

[TABLE]

The task is now to show that $\{(w_{1,n}^{k},w_{2,n}^{k})\}_{n,k}$ is relatively compact in $C_{0}^{1}(\overline{\Omega})\times C_{0}^{1}(\overline{\Omega})$. Indeed, bearing in mind that $(w_{1,n}^{k-1},w_{2,n}^{k-1})\in[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$ and $(z_{1,n},z_{2,n})\in\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C)$, on account of (3.15), one gets

[TABLE]

for every $n,\,k\in\mathbb{N}$, with $\hat{C}_{1},\,\hat{C}_{2}>0$ independent from $n$ and $k$. Applying [14, Lemma 3.1], $\{(w_{1,n}^{k},w_{2,n}^{k})\}_{n,k}$ is bounded in $C_{0}^{1,\gamma}(\overline{\Omega})\times C_{0}^{1,\gamma}(\overline{\Omega})$ and our task is achieved, being $C_{0}^{1,\gamma}(\overline{\Omega})$ compactly embedded in $C_{0}^{1}(\overline{\Omega})$. Finally, the conclusion follows by proceeding analogously to the proof of [10, Lemma 2.5, page 535]. ∎

4. Proof of the main result

According to Proposition 1, for $C>1$ large enough, for all $(z_{1},z_{2})\in\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C)$, there exists $(u^{\ast},v^{\ast})_{(z_{1},z_{2})}$ in $C_{0}^{1,\gamma}(\overline{\Omega})\times C_{0}^{1,\gamma}(\overline{\Omega})$ for certain $\gamma\in(0,1)$ that is the smallest solution within $[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$ for system ($P_{(z_{1},z_{2})}$). Thus, the operator

[TABLE]

is well defined and clearly the fixed points of the map $\mathcal{T}$ are solutions of problem ($P$).

Lemma 3**.**

The map $\mathcal{T}$ is continuous and compact.

Proof.

First, observe that $\mathcal{T}$ is compact, namely, taking in mind that $\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C)$ is bounded with respect to the $(C_{0}^{1}(\overline{\Omega})\times C_{0}^{1}(\overline{\Omega}))$-topology, for any sequence $\{(z_{1,n},z_{2,n})\}_{n}$ in $\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C)$ one has that $(u_{n}^{\ast},v_{n}^{\ast})=\mathcal{T}(z_{1,n},z_{2,n})$ is relatively compact in $C_{0}^{1}(\overline{\Omega})\times C_{0}^{1}(\overline{\Omega})$. This follows readily from (3.11) and (3.12), since $(\underline{u},\underline{v})\leq(u_{n}^{\ast},v_{n}^{\ast})\leq(\overline{u},\overline{v})$ in $\Omega$, Indeed, as in the proof of Lemma 2, applying [14, Lemma 3.1] one has that $\{(u_{n},v_{n})\}$ is bounded in $C_{0}^{1,\gamma}(\overline{\Omega})\times C_{0}^{1,\gamma}(\overline{\Omega})$ and we can conclude again invoking the compactness of the embedding $C_{0}^{1,\gamma}(\overline{\Omega})\hookrightarrow C_{0}^{1}(\overline{\Omega})$.

Let us show that $\mathcal{T}$ is continuous. Let $(z_{1,n},z_{2,n})\rightarrow(z_{1},z_{2})$ in $\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C)$ and put $(u_{n}^{*},v_{n}^{*})=\mathcal{T}(z_{1,n},z_{2,n})$. Then, we already know that there exist a subsequence $\{(u_{n_{k}},v_{n_{k}})\}_{k}$ and an element $(u_{n}^{*},v_{n}^{*})\in[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$ such that

[TABLE]

Passing to the limit in the equations

[TABLE]

[TABLE]

one gets that $(u^{*},v^{*})\in S_{(z_{1},z_{2})}$.

The proof is completed by showing that $(u^{\ast},v^{\ast})$ is the smallest solution of ($P_{(z_{1},z_{2})}$) within $[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$. Indeed, fix a solution $(w_{1},w_{2})$ of ($P_{(z_{1},z_{2})}$) such that $(w_{1},w_{2})\in[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$. We can conclude verifying that

[TABLE]

According to Lemma 2, there exists $(w_{1,n},w_{2,n})\in S_{(z_{1,n},z_{2,n})}$ such that

[TABLE]

Then, since $(u_{n_{k}}^{\ast},v_{n_{k}}^{\ast})$ is the smallest solution in $[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$ of $(P_{(z_{1,n_{k}},z_{2,n_{k}})})$, it is clear that

[TABLE]

for all $k\in\mathbb{N}$. Passing to the limit in the previous inequalities and bearing in mind (4.1) and (4.3) one directly achieves (4.2). This ends the proof. ∎

Lemma 4**.**

$\mathcal{T}(\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C))\subset\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C)$* provided $C>1$ is large enough.*

Proof.

For any $(z_{1},z_{2})\in\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C)$, let $(u^{\ast},v^{\ast})_{(z_{1},z_{2})}=\mathcal{T}(z_{1},z_{2})$ be the smallest solution of ($P_{(z_{1},z_{2})}$) in $[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$. Then, according to $\mathrm{H}(f)$, $\mathrm{H}(g)$, (3.3), (3.2) and (2.2), we get

[TABLE]

a.e. in $\Omega$ and, analogously,

[TABLE]

a.e. in $\Omega$, provided that $C>1$ is sufficiently large. Then, using the inequality in (3.6), it follows that

[TABLE]

namely $(u^{\ast},v^{\ast})_{(z_{1},z_{2})}\in\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C)$. This ends the proof of lemma. ∎

Now we are in position to prove our main result.

Proof of Theorem 1.

On the basis of Lemmas 3 and 4, Schauder’s fixed point theorem (see, e.g., [27, p. 57]) garantees the existence of $(u,v)\in\mathcal{K}(C)_{1}\times\mathcal{K}_{2}(C)$ satisfying $(u,v)=\mathcal{T}(u,v).$ Taking into account the definition of $\mathcal{T}$, it turns out that $(u,v)\in C_{0}^{1}(\overline{\Omega})\times C_{0}^{1}(\overline{\Omega})$ is a (positive) solution of problem ($P$). Since $(u,v)\in\mathcal{K}_{1}(C)\times\mathcal{K}_{2}(C)$, in particular, $(u,v)\in[\underline{u},\overline{u}]\times[\underline{v},\overline{v}]$ and on account of (3.3), (3.2) and (2.2), the property (1.1) is fullfield. This completes the proof. ∎

Acknowledgements 1**.**

This work was partially performed when the third-named author visited Reggio Calabria University, to which he is grateful for the kind hospitality.