article

Title: Critical and subcritical elliptic systems in dimension two

Authors: Djairo G. de Figueiredo, Joao Marcos do O and Bernhard Ruf

Issue: Volume 53 (2004), Issue 4, 1037-1054

Abstract:

$In this paper we study the existence of nontrivial solutions for the following system of two coupled semilinear Poisson equations: \[ \begin{cases} -\Delta u=g(v),\ v>0&\mbox{in }\Omega,\\ -\Delta v=f(u),\ u>0&\mbox{in }\Omega,\\ u=0,\ v=0,&\mbox{on }\partial\Omega, \end{cases}\tag{S} \] where $\Omega$ is a bounded domain in $\mathbb{R}^2$ with smooth boundary $\partial\Omega$, and the functions $f$ and $g$ have the maximal growth which allow us to treat problem (S) variationally in the Sobolev space $H_0^1(\Omega)$. We consider the case with nonlinearities in the critical growth range suggested by the so-called Trudinger-Moser inequality.$