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arXiv · 2406.18338

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

Achraf El wazna·Azeddine Baalal
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Abstract

In this paper, we consider the existence of solutions of the following nonhomogeneous fractional $p(x,.)$-Laplacian Dirichlet problem: \begin{equation*} \left\{\begin{aligned} \Big(-Δ_{p(x,.)}\Big)^s u (x)&=f(x, u) &\text { in }& Ω, u &=g &\text { in }& \mathbb{R}^N \setminusΩ, \end{aligned}\right. \end{equation*} where $Ω\subset\mathbb{R}^N$ is a smooth bounded domain, $\Big(-Δ_{p(x,.)}\Big)^s$ 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 $p(x, \cdot)$-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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Achraf El wazna, Azeddine Baalal. 2024-06-26. Existence of weak solutions for fractional $p(x, .)$-Laplacian Dirichlet problems with nonhomogeneous boundary conditions. https://arxiv.org/abs/2406.18338

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