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

Abstract

This paper is devoted to the study of the nonhomogeneous problem −div (a(|∇u|)) is a continuous function, a is mapping such that ϕ(|t|)t is increasing homeomorphism from R to R and g : Ω × R → R is a continuous function. We establish there main results with various assumptions, the first one asserts that any λ > 0 is an eigenvalue of our problem. The second Theorem states the existence of a constant λ * such that every λ ∈ (0, λ * ) is an eigenvalue of the problem. While the third Theorem claims the existence of a constant λ * * such that every λ ∈ [λ * * , ∞) is an eigenvalue of the problem. Our approach relies on adequate variational methods in Orlicz-Sobolev spaces.