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

Fuzzy Optimization and Modeling Journal

مقاله 5، دوره 3، شماره 1، فروردین 2022، صفحه 64-71   

نوع مقاله: Original Article

شناسه دیجیتال (DOI): 10.30495/fomj.2022.1950685.1057

نویسندگان

Asieh Rezvani¹؛ Mohsen Alimohammady^{* 2}؛ Bahram Agheli³

¹Department of Mathematics, Qaemshahr Branch, Islamic Azad university, Qaemshahr, Iran.

²Department of Mathematics, University of Mazandaran, Babolsar, Iran

³Department of Mathematics, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran.

چکیده

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.

کلیدواژه‌ها

𝑝(𝑥) – Laplacian؛ Variational Method؛ Genus Theory؛ Sobolev Space