Interior and boundary regularity results for strongly nonhomogeneous $p,q$-fractional problems

Abstract

In this article, we deal with the global regularity of weak solutions to a class of problems involving the fractional $(p,q)$-Laplacian, denoted by $(-\hat{I}”{)}_p^{s_1}+(-\hat{I}”{)}_q^{s_2}$, for $s_2,s_1\in (0,1)$ and $1<p,q<\infty$. We establish completely new Hölder continuity results, up to the boundary, for the weak solutions to fractional $(p,q)$-problems involving singular as well as regular nonlinearities. Moreover, as applications to boundary estimates, we establish new Hopf type maximum principle and strong comparison principle in both situations.

43 pages