An Ambrosetti-Prodi type result for integral equations involving dispersal operator

Abstract

In this paper we study the existence of solution for the following class of nonlocal problems \[ L_0u =f(x,u)+g(x) , \ \mbox{in} \ Ω, \] where $\Omega \subset {\mathbb{R}}^N$, $N\ge 1$, is a bounded connected open, $g\in C(\overline{\Omega })$, $f:\overline{\Omega }\times \mathbb{R}\to \mathbb{R}$ are function, and $L_0:C(\overline{\Omega })\to C(\overline{\Omega })$ is a nonlocal dispersal operator. Using a sub-supersolution method and the degree theory for $\gamma$-Condensing maps, we have obtained a result of the Ambrosetti-Prodi type, that is, we obtain a necessary condition on $g$ for the non-existence of solutions, the existence of at least one solution, and the existence of at least two distinct solutions.