Regularity for mixed-order nonlinear fractional equations with degenerate coefficients

TL;DR

This paper establishes regularity results, including local boundedness and H"older continuity, for solutions to a class of mixed-order nonlinear fractional equations with degenerate coefficients, extending understanding of their qualitative behavior.

Contribution

It introduces new regularity results for mixed-order nonlinear fractional equations with degenerate coefficients, including H"older continuity and Harnack inequality under natural assumptions.

Findings

Proved local boundedness of weak solutions.

Established H"older regularity of solutions.

Derived Harnack inequality when coefficients are constant.

Abstract

We consider a class of nonlinear integro-differential equations whose leading operator is obtained as a superposition of $(-\Delta_{p})^{s}$ and $(-\Delta_{p})^{t}$, 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 H\"older 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.