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

In this paper, we study the existence of solution for the following class of nonlocal problem, $$\{\begin{array}{l}-\mathrm{\Delta }u=(\lambda f(x)-{\int }_{{\mathbb{R}}^N}K(x,y)\mathrm{\mid }u(y){\mathrm{\mid }}^{\gamma }dy)u,in{\mathbb{R}}^N,\\ {\displaystyle \underset{\mathrm{\mid }x\mathrm{\mid }\to +\mathrm{\infty }}{\lim }u(x)=0,u>0\text{in}{\mathbb{R}}^N,}\end{array}\text{eqno}(P)$$ where $N\ge 3$, $\lambda >0, \gamma\in[1,2)$,$f:\R\rightarrow\R$isapositivecontinuousfunctionand$K:\R^N\times\R^N\rightarrow\R$isanonnegativefunction\mathrm{.}Thefunctions$f$and$K$ satisfy some conditions, which permit to use Bifurcation Theory to prove the existence of solution for problem $(P)$.