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Mathematics   > Home   > Advances in Pure and Applied Mathematics   > Issue 4 (September 2021)   > Article

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

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


NAWAL IRZI
University of Tunis El Manar
Tunisia



Published on 6 September 2021   DOI : 10.21494/ISTE.OP.2021.0722

Abstract

Résumé

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This paper is devoted to the study of the nonhomogeneous problem
div(a(|u|)u)+a(|u|)u=λV(x)|u|m(x)2uμg(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¯× is a continuous function. We establish there main results with various assumptions, the first one asserts that any λ0> 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)2uμg(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¯× is a continuous function. We establish there main results with various assumptions, the first one asserts that any λ0> 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.

Mountain pass Theorem Ekeland’s variational principle Orlicz-Sobolev space.

Ekeland’s variational principle Mountain pass Theorem Orlicz-Sobolev space.

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