Sobolev versus Hölder minimizers for the degenerate fractional $p$-Laplacian

Abstract

We consider a nonlinear pseudo-differential equation driven by the fractional $p$-Laplacian $(-\Delta {)}_p^s$ with $s\in (0,1)$ and $p\ge 2$ (degenerate case), under Dirichlet type conditions in a smooth domain $\Omega$. We prove that local minimizers of the associated energy functional in the fractional Sobolev space $W_0^{s,p}(\Omega )$ and in the weighted Hölder space $C_s^0(\overline{\Omega })$, respectively, do coincide.

14 pages