Multiplicity of Solutions for a Class of Quasilinear Elliptic Systems in Orlicz-Sobolev Spaces

Abstract

In this paper, we investigate the following nonlinear and non-homogeneous elliptic system \[ \begin{cases} -\operatorname{div}(a_1(|\nabla u|) \nabla u) = \lambda_1 F_u(x,u,v) - \lambda_2 G_u(x,u,v) - \lambda_3 H_u(x,u,v) &\textrm{in $\Omega$}, \\ -\operatorname{div}(a_2(|\nabla v|) \nabla v) = \lambda_1 F_v(x,u,v) - \lambda_2 G_v(x,u,v) - \lambda_3 H_v(x,u,v) &\textrm{in $\Omega$}, \\ u = v = 0 &\textrm{on $\partial \Omega$}, \end{cases} \] where $\Omega$ is a bounded domain in $\mathbb{R}^N$ ($N \geq 1$) with smooth boundary $\partial \Omega$, $\lambda_1$, $\lambda_2$, $\lambda_3$ are three parameters, $\phi_i(t) = a_i(|t|)t$ ($i = 1,2$) are two increasing homeomorphisms from $\mathbb{R}$ onto $\mathbb{R}$, and functions $F$, $G$, $H$ are of class $C^1(\Omega \times \mathbb{R}^2, \mathbb{R})$ and satisfy some reasonable growth conditions. By using a three critical points theorem due to B. Ricceri, we obtain that system has at least three solutions. With some additional conditions, by using a four critical points theorem due to G. Anello, we obtain that system has at least four solutions.