Critical and Subcritical Elliptic Systems in dimension two

In this paper we study the existence of nontrivial solutions for the following system of two coulped semilinear Poisson equations:\[\left{\begin{array}{rclccl}-\Delta u&=&g(v),&v>0&in &\Omega,\\-\Delta v&=&f(u),&u>0&in&\Omega,\\u &=&0, &v=0&on&\partial\Omega,\\\end{array}\right.\]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.