Existence and Multiplicity Results for the p(x)- Laplacian Equation via Genus Theory

Subject Areas : Fuzzy Optimization and Modeling Journal

Asieh Rezvani ¹ , Mohsen Alimohammady ² , Bahram Agheli ³

1 - Department of Mathematics, Qaemshahr Branch, Islamic Azad university, Qaemshahr, Iran.
2 - Department of Mathematics, University of Mazandaran, Babolsar, Iran
3 - Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran.

Received: 2022-01-25 Accepted : 2022-03-11 Published : 2022-01-01

Keywords: Variational method, genus theory, Sobolev space, 𝑝(𝑥) – Laplacian,

Abstract :

In this paper, we study existence and multiplicity of nontrivial weak solutions for the following equation involving weight and variable exponents −𝑑𝑖𝑣 (1+|∇𝑢|2)𝑝(𝑥)−22∇𝑢=𝜆𝑚(𝑥)|𝑢|𝑝(𝑥)−2𝑢, 𝑖𝑛Ω, where Ω is a bounded domain of ℝ𝑁 with smooth enough boundary which is subject to Dirichlet boundary condition, 𝜆 is a positive real parameter and 𝑝 is real continuos function on Ω̅ with 1<𝑝(𝑥)<𝑝∗(𝑥), where 𝑝∗(𝑥)=𝑁𝑝(𝑥)𝑁−𝑝(𝑥) and 𝑝(𝑥)<𝑁 for all 𝑥∈Ω̅ , 𝑚:Ω̅→[0,∞) is a continuous function. By using variational method and Krasnoselskii,s genus theory, we show the existence and multiplicity of the solutions. For this purpose we work on a generalized variable exponent Lebesgue-Sobolev space.