Regularity for mixed-order nonlinear fractional equations with degenerate coefficients

Abstract

We consider a class of nonlinear integro-differential equations whose leading operator is obtained as a superposition of $(-\hat{I}{”}_p{)}^s$ and $(-\hat{I}{”}_p{)}^t$, where $0<s<t<1<p<\infty$, weighted via two possibly degenerate coefficients $a(\cdot ,\cdot ),b(\cdot ,\cdot )\ge 0$. We prove local boundedness and Hölder regularity of its weak solutions under natural assumptions on the coefficients $a(\cdot ,\cdot )$, $b(\cdot ,\cdot )$ and the powers $s,t$, and $p$. Moreover, when $a(\cdot ,\cdot )\equiv 1$, we also prove a Harnack inequality for weak solutions.