We prove that for p ≥ 2 solutions of equations modeled by the fractional p−Laplacian improve their regularity on the scale of fractional Sobolev spaces. Moreover, under certain precise conditions, they are in W 1,p loc and their gradients are in a fractional Sobolev space as well. The relevant estimates are stable as the fractional order of differentiation s reaches 1.

In this paper we give sufficient conditions on the approximating domains in order to obtain the continuity of solutions for the fractional p−laplacian. These conditions are given in terms of the fractional capacity of the approximating domains.

The main purpose of this paper was to consider new sandwich pairs and investigate the existence a solution for class fractional differential equations with p-Laplacian via variational methods in ψ-fractional space Hpα,β;ψ(Ω). results obtained are first make use theory ψ-Hilfer operators p-Laplacian.