Existence of solution for a nonlocal dispersal model with nonlocal term via bifurcation theory

Abstract

In this paper we study the existence of solution for the following class of nonlocal problems \[ L_0u =u \left(λ- \int_ΩQ(x,y) |u(y)|^p dy \right) , \ \mbox{in} \ Ω, \] where $\Omega \subset {\mathbb{R}}^N$, $N\ge 1$, is a bounded connected open, $p>0$, $\lambda$ is a real parameter, $Q:\Omega \times \Omega \to \mathbb{R}$ is a nonnegative function, and $L_0:C(\overline{\Omega })\to (\overline{\Omega })$ is a nonlocal dispersal operator. The existence of solution is obtained via bifurcation theory.

In this version, we correct some mistakes in the old version, for example, in the old version was used that the inverse of operator L_0+MI is compact, which is not true. The reader will see that we assumed some additional conditions on function $Q$