@ARTICLE{10.21494/ISTE.OP.2021.0722, TITLE={Multiplicity of solutions for a nonhomogeneous problem involving a potential in Orlicz-Sobolev spaces}, AUTHOR={NAWAL IRZI, }, JOURNAL={Advances in Pure and Applied Mathematics}, VOLUME={12}, NUMBER={Issue 4 (September 2021)}, YEAR={2021}, URL={https://www.openscience.fr/Multiplicity-of-solutions-for-a-nonhomogeneous-problem-involving-a-potential-in}, DOI={10.21494/ISTE.OP.2021.0722}, ISSN={1869-6090}, ABSTRACT={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.}}