Global regularity results for non-homogeneous growth fractional problems

TL;DR

This paper proves global H"older regularity for solutions to fractional $(p,q)$-Laplacian problems, extending previous results to subquadratic cases and more general right-hand sides using new techniques.

Contribution

It introduces a novel approach to establish global regularity for fractional $(p,q)$-Laplacian problems, including subquadratic cases and general right-hand sides.

Findings

Proved global H"older regularity for weak solutions.

Established a nonlocal Harnack inequality for solutions.

Extended regularity results to the subquadratic case ($q<2$).

Abstract

This article concerns with the global H\"older regularity of weak solutions to a class of problems involving the fractional $(p,q)$-Laplacian, denoted by $(-\Delta)^{s_1}_{p}+(-\Delta)^{s_2}_{q}$, for $1<p,q<\infty$ and $s_1,s_2\in (0,1)$. We use a suitable Caccioppoli inequality and local boundedness result in order to prove the weak Harnack type inequality. Consequently, by employing a suitable iteration process, we establish the interior H\"older regularity for local weak solutions, which need not be assumed bounded. The global H\"older regularity result we prove expands and improves the regularity results of Giacomoni, Kumar and Sreenadh (arXiv: 2102.06080) to the subquadratic case (that is, $q<2$) and more general right hand side, which requires a different and new approach. Moreover, we establish a nonlocal Harnack type inequality for weak solutions, which is of independent interest.