On unbounded solutions of ergodic problems for non-local Hamilton-Jacobi equations

TL;DR

This paper investigates the existence of unbounded solutions to a non-local Hamilton-Jacobi ergodic problem, revealing a threshold growth condition for the data function and identifying a critical ergodic constant that determines solution existence.

Contribution

It introduces a novel threshold growth criterion for solutions and characterizes the critical ergodic constant in non-local Hamilton-Jacobi equations, unlike the local case.

Findings

Existence of a threshold growth for the function f that separates solution existence and non-existence.

Existence of a critical ergodic constant λ* with unique solutions for λ ≤ λ*.

Solutions are unique up to an additive constant and only bounded from below.

Abstract

We study an ergodic problem associated to a non-local Hamilton-Jacobi equation defined on the whole space $\lambda-\mathcal{L}[u](x)+|Du(x)|^m=f(x)$ and determine whether (unbounded) solutions exist or not. We prove that there is a threshold growth of the function $f$, that separates existence and non-existence of solutions, a {phenomenon} that does not appear in the local version of the problem. Moreover, we show that there exists a critical ergodic constant, $\lambda_*$, such that the ergodic problem has solutions for $\lambda\leq \lambda_*$ and such that the only solution bounded from below, which is unique up to an additive constant, is the one associated to $\lambda_*$.