Existence of solution for a nonlocal problem in $\R^N$ via bifurcation theory

TL;DR

This paper proves the existence of solutions for a class of nonlocal elliptic problems in  space using bifurcation theory, under certain conditions on the involved functions.

Contribution

It introduces a novel application of bifurcation theory to establish solutions for nonlocal problems with integral terms in  space.

Findings

Existence of positive solutions under specified conditions.

Application of bifurcation theory to nonlocal PDEs.

Conditions on functions f and K enable solution existence.

Abstract

In this paper, we study the existence of solution for the following class of nonlocal problem, $$ \left\{ \begin{array}{lcl} -\Delta u=\left(\lambda f(x)-\int_{\R^N}K(x,y)|u(y)|^{\gamma}dy\right)u,\quad \mbox{in} \quad \R^{N}, \\ \displaystyle \lim_{|x| \to +\infty}u(x)=0,\quad u>0 \quad \text{in} \quad \R^{N}, \end{array} \right. \eqno{(P)} $$ where $N\geq3$, $\lambda >0, \gamma\in[1,2)$, $f:\R\rightarrow\R$ is a positive continuous function and $K:\R^N\times\R^N\rightarrow\R$ is a nonnegative function. The functions $f$ and $K$ satisfy some conditions, which permit to use Bifurcation Theory to prove the existence of solution for problem $(P)$.