Existence of nontrivial solution for elliptic systems involving the p(x)-Laplacian

Abstract

In this paper, we consider the system \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left\{ {\begin{array}{*{20}c} {\left\{ { - \Delta _{p\left( x \right)} u = \lambda a\left( x \right)\left| u \right|} \right.^{r_1 \left( x \right) - 2} u - \mu b\left( x \right)\left| u \right|^{\alpha \left( x \right) - 2} u\;x \in \Omega } \\ {\left\{ { - \Delta _{q\left( x \right)} \nu = \lambda c\left( x \right)\left| \nu \right|} \right.^{r_2 \left( x \right) - 2} \nu - \mu d\left( x \right)\left| \nu \right|^{\beta \left( x \right) - 2} \nu \;x \in \Omega } \\ {u = \nu = 0\;x \in \partial \Omega } \\ \end{array} } \right.$$ \end{document} where Ω is a bounded domain in ℝN with smooth boundary, λ, μ > 0, p, q, r1, r2, α and β are continuous functions on \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\bar \Omega$$ \end{document} satisfying appropriate conditions. We prove that for any μ > 0, there exists λ* sufficiently small, and λ* large enough such that for any λ ∈ (0; λ*) ∪ (λ*, ∞), the above system has a nontrivial weak solution. The proof relies on some variational arguments based on the Ekeland’s variational principle and some adequate variational methods.