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. پرونده مقاله

In this paper, we establish the existence and uniqueness result of the linearSchrodinger equation with Marchaud fractional derivative in Colombeau generalized algebra.The purpose of introducing Marchaud fractional derivative is regularizing it in Colombeausense. پرونده مقاله