Continuity of weak solutions to an elliptic problem on $p$-fractional Laplacian

Abstract

In this paper we study an elliptic variational problem regarding the $p$-fractional Laplacian in ${\mathbb{R}}^N$ on the basis of recent result \cite{Ha1}, which generalizes the nice work \cite{AT,AP,XZR1}, and then give some sufficient conditions under which some weak solutions to the above elliptic variational problem are continuous in ${\mathbb{R}}^N$. In the final appendix we correct the proofs of both \cite[Lemma 10]{PXZ1} and \cite[Lemma A.6]{PXZ} for $1<p<2$.