Existence of positive solution for a system of elliptic equations via bifurcation theory

Abstract

In this paper we study the existence of solution for the following class of system of elliptic equations $$ \left\{ \begin{array}{lcl} -Δu=\left(a-\int_ΩK(x,y)f(u,v)dy\right)u+bv,\quad \mbox{in} \quad Ω -Δv=\left(d-\int_ΩΓ(x,y)g(u,v)dy\right)v+cu,\quad \mbox{in} \quad Ωu=v=0,\quad \mbox{on} \quad \partialΩ \end{array} \right. \eqno{(P)} $$ where $\Omega \subset {\mathbb{R}}^N$ is a smooth bounded domain, $N\ge 1$, and $K,\Gamma :\Omega \times \Omega \to \mathbb{R}$ is a nonnegative function checking some hypotheses and $a,b,c,d\in \mathbb{R}$. The functions $f$ and $g$ satisfy some conditions which permit to use Bifurcation Theory to prove the existence of solution for $(P)$.

21 pages. arXiv admin note: text overlap with arXiv:1509.05294