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Strong maximum principle for fully nonlinear nonlocal problems

Juan Pablo Cabeza, Gabrielle Nornberg, Disson dos Prazeres

TL;DR

This paper analyzes the strong maximum principle (SMP) and dead-core formation for fully nonlinear, nonlocal equations of the form $\mathcal{M}^{\pm}[u]+a(x)u^{q}=0$ with $q\in(0,1)$ on a bounded convex domain $\Omega$. It develops a viscosity-solution framework, proves a nonlocal Hopf lemma, and derives a quantitative SMP under a smallness condition on the negative part of the solution, employing Liouville-type arguments and barrier constructions. The authors also establish existence of nontrivial solutions through eigenfunction-based sub- and supersolutions and demonstrate a counterexample where SMP fails when the exterior negative part is large. These results advance understanding of dead-core phenomena and boundary behavior for fully nonlinear nonlocal problems, with potential applications to reaction-diffusion and extinction dynamics.

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 $$ \mathcal{M}^{\pm}[u]+a(x)u^{q}(x)=0 \; \text{ in }Ω,\qquad u\geq 0 \; \text{ in }Ω. $$ Here $Ω\subset \mathbb{R}^n$ 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.

Strong maximum principle for fully nonlinear nonlocal problems

TL;DR

This paper analyzes the strong maximum principle (SMP) and dead-core formation for fully nonlinear, nonlocal equations of the form with on a bounded convex domain . It develops a viscosity-solution framework, proves a nonlocal Hopf lemma, and derives a quantitative SMP under a smallness condition on the negative part of the solution, employing Liouville-type arguments and barrier constructions. The authors also establish existence of nontrivial solutions through eigenfunction-based sub- and supersolutions and demonstrate a counterexample where SMP fails when the exterior negative part is large. These results advance understanding of dead-core phenomena and boundary behavior for fully nonlinear nonlocal problems, with potential applications to reaction-diffusion and extinction dynamics.

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 Here is a bounded convex domain, stands for nonlocal Pucci extremal operators defined in a class of homogeneous kernels, , and 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.
Paper Structure (5 sections, 14 theorems, 91 equations)

This paper contains 5 sections, 14 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Auxiliary tools
  3. Hopf Lemma and Strong Maximum Principle
  4. Strong Maximum principle for sublinear equations
  5. Existence of nontrivial solutions and a solution that violates SMP

Key Result

Theorem 1

Let $\Omega$ be a bounded $C^{1,1}$ convex domain and $q\in (0,1)$. Let $a\in C(\overline{\Omega})$ such that $a^+\not\equiv 0$. Then there exists $\delta > 0$ so that if and then every nontrivial solution of eq:puccinonlocal such that $u \leq 0$ in $\mathbb{R}^{n}\backslash\Omega$ belongs to $\mathcal{P}^{0}(\Omega)$.

Theorems & Definitions (31)

  • Theorem 1
  • Theorem 2
  • Definition 1
  • Lemma 1
  • Lemma 2
  • proof
  • Proposition 1
  • Theorem 3
  • proof
  • Proposition 2: Strong Maximum Principle
  • ...and 21 more