On boundary regularity for the fractional p-Laplacian with unbounded reactions

Abstract

We consider an elliptic equation driven by the $s$-fractional $p$-Laplacian, set in a smooth bounded domain $\Omega \subset {\mathbb{R}}^N$ with homogeneous nonlocal Dirichlet conditions and a reaction $f$ lying in $L^q(\Omega )$ for some $q\ge 1$. We prove that the unique solution $u$ is $\alpha$-Hölder continuous up to the boundary, for any $\alpha$ below $p^{\prime }(s-N/pq)$ if $N/ps<q\le N/s$, and $\alpha =s$ if $q>N/s$. Also, we prove that if $q>N/s$ then $u/{\mathrm{d}}_{\Omega }^s$ admits a Hölder continuous extension to the closure of $\Omega$, where ${\mathrm{d}}_{\Omega }$ denotes the distance from the boundary. Our results are almost optimal and extend previous regularity theorems known in the linear case.

37 pages, 3 figures