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

TL;DR

This paper investigates the continuity of weak solutions to a variational elliptic problem involving the $p$-fractional Laplacian in $ abla^N$, building on recent results and providing conditions for solution continuity.

Contribution

It extends previous work by establishing new sufficient conditions for the continuity of weak solutions to the $p$-fractional Laplacian problem.

Findings

Identifies conditions ensuring solution continuity in $ abla^N$.

Provides corrected proofs for key lemmas in related literature.

Generalizes previous results on fractional Laplacian problems.

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 $1<p<2$.