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

TL;DR

This paper proves the existence of weak solutions for a class of fractional p(x,.)-Laplacian problems with nonhomogeneous boundary conditions, using variable exponent Sobolev spaces and fixed point theory.

Contribution

It establishes the existence of solutions for fractional p(x,.)-Laplacian problems with nonhomogeneous boundary data, extending previous results to variable exponent and fractional settings.

Findings

Existence of weak solutions under certain conditions.

Application of Schauder's fixed point theorem in variable exponent spaces.

Development of a framework for fractional p(x,.)-Laplacian with nonhomogeneous boundaries.

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(-\Delta_{p(x,.)}\Big)^s u (x)&=f(x, u) &\text { in }& \Omega, u &=g &\text { in }& \mathbb{R}^N \setminus\Omega, \end{aligned}\right. \end{equation*} where $\Omega\subset\mathbb{R}^N$ is a smooth bounded domain, $\Big(-\Delta_{p(x,.)}\Big)^s$ is the fractional $p(x,.)$-Laplacian, $f$ is a Carath\'eodory 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.