Existence of weak solutions for fractional p(x, .)-Laplacian Dirichlet problems with nonhomogeneous boundary conditions

2024/06/26 by wazna, Achraf El, Baalal, Azeddine
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2406.18338

Abstract

In this paper, we consider the existence of solutions of the following nonhomogeneous fractional p(x,.)-Laplacian Dirichlet problem: \\beginaligned (-Δ_p(x,.))^s u (x)amp;=f(x, u) amp;\text in amp; Ω, u amp;=g amp;\text in amp; ℝ^N ∖Ω, \endaligned. where Ω⊂ℝ^N is a smooth bounded domain, (-Δ_p(x,.))^s is the fractional p(x,.)-Laplacian, f is a Carathéodory function with suitable growth condition and g is a given boundary data. The proof of our main existence results relies on the study of the fractional p(x, ⋅)-Poisson equation with a nonhomogeneous Dirichlet boundary condition and the theory of fractional Sobolev spaces with variable exponents, together with Schauder's fixed point theorem.

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