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

In this paper, we consider the existence of solutions of the following nonhomogeneous fractional p (x,. ) -Laplacian Dirichlet problem: equation* \aligned (-₏ (ₗ,. ) ) ˢ u (x) &=f (x, u) & { in &, u &=g & in & RN, aligned. equation* where N is a smooth bounded domain, (-₏ (ₗ,. ) ) 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, ) -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.