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

Abstract

In this paper, we study the existence of solution for the following class of nonlocal problem, $$ \left\{ \begin{array}{lcl} -Δu=\left(λf(x)-\int_{\R^N}K(x,y)|u(y)|^γdy\right)u,\quad \mbox{in} \quad \R^{N}, \\ \displaystyle \lim_{|x| \to +\infty}u(x)=0,\quad u>0 \quad \text{in} \quad \R^{N}, \end{array} \right. \eqno{(P)} $$ where $N\ge 3$, $\lambda >0,\gamma \in [1,2)$, $f:\mathbb{R}\to \mathbb{R}$ is a positive continuous function and $K:{\mathbb{R}}^N\times {\mathbb{R}}^N\to \mathbb{R}$ 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 $(P)$.

References in corpus (1)

• Convergence to equilibrium for positive solutions of some mutation-selection model