## Taiwanese Journal of Mathematics

### 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.

#### Article information

Source
Taiwanese J. Math., Volume 21, Number 4 (2017), 881-912.

Dates
Received: 4 September 2016
Revised: 24 November 2016
Accepted: 27 November 2016
First available in Project Euclid: 27 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1501120840

Digital Object Identifier
doi:10.11650/tjm/7887

Mathematical Reviews number (MathSciNet)
MR3684392

Zentralblatt MATH identifier
06871351

#### Citation

Wang, Liben; Zhang, Xingyong; Fang, Hui. Multiplicity of Solutions for a Class of Quasilinear Elliptic Systems in Orlicz-Sobolev Spaces. Taiwanese J. Math. 21 (2017), no. 4, 881--912. doi:10.11650/tjm/7887. https://projecteuclid.org/euclid.twjm/1501120840