# Multiplicité de solutions pour un problème non homogène impliquant un potentiel dans les espaces d’Orlicz-Sobolev

## Multiplicity of solutions for a nonhomogeneous problem involving a potential in Orlicz-Sobolev spaces

NAWAL IRZI
University of Tunis El Manar
Tunisia

Publié le 6 septembre 2021   DOI : 10.21494/ISTE.OP.2021.0722

### Keywords

This paper is devoted to the study of the nonhomogeneous problem
$-div (a(|\nabla u|)\nabla u)+a(| u|)u=\lambda V(x)|u|^{m(x)-2}u-\mu g(x,u) \mbox{ in} \ \Omega, \ u=0 \mbox{ on} \ \partial\Omega ,$ where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N,\lambda, \mu$ are positive real numbers, $V(x)$ is a potential, $m: \overline{ \Omega} \to (1, \infty)$ is a continuous function, $a$ is mapping such that $\varphi(|t|)t$ is increasing homeomorphism from ℝ to ℝ and $g: \overline{\Omega}\times ℝ \to ℝ$ is a continuous function. We establish there main results with various assumptions, the first one asserts that any $\lambda$0> is an eigenvalue of our problem. The second Theorem states the existence of a constant $\lambda^{*}$ such that every $\lambda \in (0,\lambda^{*})$ is an eigenvalue of the problem. While the third Theorem claims the existence of a constant $\lambda^{**}$ such that every $\lambda \in [\lambda^{**},\infty)$ is an eigenvalue of the problem. Our approach relies on adequate variational methods in Orlicz-Sobolev spaces.

This paper is devoted to the study of the nonhomogeneous problem
$-div (a(|\nabla u|)\nabla u)+a(| u|)u=\lambda V(x)|u|^{m(x)-2}u-\mu g(x,u) \mbox{ in} \ \Omega, \ u=0 \mbox{ on} \ \partial\Omega ,$ where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N,\lambda, \mu$ are positive real numbers, $V(x)$ is a potential, $m: \overline{ \Omega} \to (1, \infty)$ is a continuous function, $a$ is mapping such that $\varphi(|t|)t$ is increasing homeomorphism from ℝ to ℝ and $g: \overline{\Omega}\times ℝ \to ℝ$ is a continuous function. We establish there main results with various assumptions, the first one asserts that any $\lambda$0> is an eigenvalue of our problem. The second Theorem states the existence of a constant $\lambda^{*}$ such that every $\lambda \in (0,\lambda^{*})$ is an eigenvalue of the problem. While the third Theorem claims the existence of a constant $\lambda^{**}$ such that every $\lambda \in [\lambda^{**},\infty)$ is an eigenvalue of the problem. Our approach relies on adequate variational methods in Orlicz-Sobolev spaces.