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

In this paper, we consider the existence of solutions of the following nonhomogeneous fractional $p(x,\mathrm{.})$-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 $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a smooth bounded domain, $(-{\mathrm{\Delta }}_{p(x,\mathrm{.})}{)}^s$ is the fractional $p(x,\mathrm{.})$-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.