On fractional quasilinear equations with elliptic degeneracy

TL;DR

This paper develops a systematic approach to study solutions of nonlocal quasilinear equations with gradient degeneracy, establishing existence, multiplicity, and regularity results using geometric and uniqueness arguments.

Contribution

It introduces a novel geometric method tailored for nonlocal equations to analyze solution properties, including regularity and multiplicity, with a focus on gradient degeneracy.

Findings

Existence and multiplicity of solutions demonstrated for certain classes of exterior data.

Gradient Hölder regularity estimates established for solutions.

Identification of cases where uniqueness of solutions is confirmed.

Abstract

In this work, we present a systematic approach to investigate the existence, multiplicity, and local gradient regularity of solutions for nonlocal quasilinear equations with local gradient degeneracy. Our method involves an interactive geometric argument that interplays with uniqueness property for the corresponding homogeneous problem, leading with gradient H\"older regularity estimates. This approach is intrinsically developed for nonlocal scenarios, where uniqueness holds for the local homogeneous problem. We illustrate our results by showing classes of exterior data that exhibit multiple solutions, while also highlighting relevant cases where uniqueness is confirmed.