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

Abstract

We study an ergodic problem associated to a non-local Hamilton–Jacobi equation defined on the whole space λ−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, λ∗, such that the ergodic problem has solutions for λ⩽λ∗ and such that the only solution bounded from below, which is unique up to an additive constant, is the one associated to λ∗.