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

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 $(P)$.