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 $\lambda -\mathcal{L}[u](x)+|Du(x){|}^m=f(x)$ 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, ${\lambda }_{\ast }$, such that the ergodic problem has solutions for $\lambda \le {\lambda }_{\ast }$ and such that the only solution bounded from below, which is unique up to an additive constant, is the one associated to ${\lambda }_{\ast }$.