Strong maximum principle for fully nonlinear nonlocal problems

TL;DR

This paper establishes a strong maximum principle for nonnegative solutions of fully nonlinear nonlocal equations, introducing new conditions on solutions outside the domain and proving related Liouville theorems and a Hopf lemma.

Contribution

It introduces a novel nonlocal hypothesis affecting dead core formation and proves a Hopf lemma for viscosity solutions of fully nonlinear operators.

Findings

Established a strong maximum principle for nonlocal fully nonlinear problems.

Proved a Hopf lemma for viscosity solutions of nonlocal operators.

Identified conditions influencing dead core formation in solutions.

Abstract

In this paper, we study solvability and qualitative properties of nonnegative solutions for a sublinear nonlocal problem with fully nonlinear structure in the form $$ \mathcal{M}^{\pm}[u]+a(x)u^{q}(x)=0 \; \text{ in }\Omega,\qquad u\geq 0 \; \text{ in }\Omega. $$ Here $\Omega \subset \mathbb{R}^n$ is a bounded $C^{1,1}$ convex domain, $\mathcal{M}^{ \pm}$ stands for nonlocal Pucci extremal operators defined in a class $\mathcal{L}_*$ of homogeneous kernels, $q\in(0,1)$, and $a$ is a possibly sign-changing weight. We introduce a new nonlocal hypothesis on the negative part of the solution outside the domain, which together with the negative part of the potential, influences the formation of dead cores and cannot be removed. Our approach relies on uniform bounds from below of the maximum of nontrivial solutions through Liouville theorems, and on a Hopf lemma for viscosity solutions driven by fully nonlinear operators, which we also prove.