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

TL;DR

This paper investigates the existence of solutions for a class of nonlocal dispersal models with nonlocal terms, using bifurcation theory to establish conditions under which solutions exist.

Contribution

It introduces a novel application of bifurcation theory to prove the existence of solutions for nonlocal dispersal equations with nonlocal terms.

Findings

Existence of solutions established for the nonlocal problem.

Bifurcation theory effectively used to analyze solution existence.

Results applicable to a broad class of nonlocal dispersal models.

Abstract

In this paper we study the existence of solution for the following class of nonlocal problems \[ L_0u =u \left(\lambda - \int_{\Omega}Q(x,y) |u(y)|^p dy \right) , \ \mbox{in} \ \Omega, \] where $\Omega \subset \mathbb{R}^{N}$, $N\geq 1$, 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.