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

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

NAWAL IRZI
University of Tunis El Manar
Tunisia

Publié le 14 septembre 2020   DOI :

### Keywords

This paper is devoted to the study of the nonhomogeneous problem –div(a(|∇u|)∇u) + a(|u|)u = λV (x)|u|m(x)-2u – μg(x,u) in Ω, u = 0 on ∂Ω where Ω is a bounded smooth domain in ℝN, λ, μ are positive real numbers, V (x) is a potential, m : Ω → (1, ∞) 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.

This paper is devoted to the study of the nonhomogeneous problem –div(a(|∇u|)∇u) + a(|u|)u = λV (x)|u|m(x)-2u – μg(x,u) in Ω, u = 0 on ∂Ω where Ω is a bounded smooth domain in ℝN, λ, μ are positive real numbers, V (x) is a potential, m : Ω → (1, ∞) 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.