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

Antonio Iannizzotto, Sunra Mosconi, Marco Squassina

Published 2019-07-20Version 1

We consider a nonlinear pseudo-differential equation driven by the fractional $p$-Laplacian $(-\Delta)^s_p$ 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 $W^{s,p}_0(\Omega)$ and in the weighted H\"older space $C^0_s(\overline\Omega)$, respectively, do coincide.