Interior and boundary regularity of fractional $p,q$-problems: the subquadratic case

Jacques Giacomoni, Deepak Kumar, K. Sreenadh

Published 2021-06-04Version 1

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. The global H\"older regularity result we prove extends and complements the regularity result of Giacomoni, Kumar and Sreenadh (arXiv: 2102.06080) to the subquadratic case (that is, $q<2$). Moreover, we establish a nonlocal Harnack type inequality for weak solutions, which is of independent interest.