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 $\alpha$-H\"older continuous up to the boundary, for any $\alpha$ below $p'(s-N/pq)$ if $N/ps N/s$. Also, we prove that if $q>N/s$ then $u/{\rm d}_\Omega^s$ admits a H\"older continuous extension to the closure of $\Omega$, where ${\rm d}_\Omega$ denotes the distance from the boundary. Our results are almost optimal and extend previous regularity theorems known in the linear case.