Existence of positive solutions for variable exponent elliptic systems with multiple parameters

Abstract

In this paper, we consider the system of differential equations

$$\begin{aligned} \left\{ \begin{array}{l@{\quad }l@{\quad }l} -\Delta _{p(x)} u=\lambda ^{p(x)}[\lambda _{1}f(v)+\mu _{1}h(u)] &{} \mathrm{in }\, \Omega , \\ -\Delta _{p(x)} v=\lambda ^{p(x)}[\lambda _{2}g(u)+\mu _{2}\tau (v)] &{} \mathrm{in }\,\Omega , \\ u=v=0 &{} \mathrm{on }\,\partial \Omega ,\\ \end{array}\right. \end{aligned}$$

where $\Omega \subset \mathbb R ^N$ is a bounded domain with $C^2$ boundary $\partial \Omega$ and $1<p(x)\in C^{1}(\overline{\Omega })$ is a function. The operator $-\Delta _{p(x)} u = -\mathrm{div}(|\nabla u|^{p(x)-2}\nabla u)$ is called the $p(x)$-Laplacian, $\lambda ,\,\lambda _{1},\,\lambda _{2},\,\mu _{ 1}$ and $\mu _{2}$ are positive parameters. Using the sub-supersolutions method, we prove the existence of positive solutions if

$$\begin{aligned} \lim _{u \rightarrow +\infty }\frac{f[M(g(u))^\frac{1}{p^--1}]}{u^{p^--1}} = 0, \quad \forall M > 0, \end{aligned}$$

$$\begin{aligned} \lim _{u \rightarrow +\infty } \frac{h(u)}{u^{p^-- 1}} = 0, \quad \lim _{u \rightarrow +\infty } \frac{\tau (u)}{u^{p^- - 1}} = 0. \end{aligned}$$

In particular, we do not assume any symmetry condition, and we do not assume any sign condition on $f(0), g(0), h(0)$ or $\tau (0)$.