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

TL;DR

This paper investigates the existence of positive solutions for a class of elliptic systems using bifurcation theory, focusing on integral operators and nonlinear functions within bounded domains.

Contribution

It introduces a bifurcation approach to establish positive solutions for elliptic systems involving integral operators and nonlinearities.

Findings

Existence of solutions established under certain conditions.

Bifurcation theory effectively applied to nonlinear elliptic systems.

Results extend previous methods to systems with integral terms.

Abstract

In this paper we study the existence of solution for the following class of system of elliptic equations $$ \left\{ \begin{array}{lcl} -\Delta u=\left(a-\int_{\Omega}K(x,y)f(u,v)dy\right)u+bv,\quad \mbox{in} \quad \Omega -\Delta v=\left(d-\int_{\Omega}\Gamma(x,y)g(u,v)dy\right)v+cu,\quad \mbox{in} \quad \Omega u=v=0,\quad \mbox{on} \quad \partial\Omega \end{array} \right. \eqno{(P)} $$ where $\Omega\subset\R^N$ is a smooth bounded domain, $N\geq1$, and $K,\Gamma:\Omega\times\Omega\rightarrow\R$ is a nonnegative function checking some hypotheses and $a,b,c,d\in\R$. The functions $f$ and $g$ satisfy some conditions which permit to use Bifurcation Theory to prove the existence of solution for $(P)$.