Positive ground states for a system of Schr\"odinger equations with critically growing nonlinearities

Abstract

We study the following problem \[ \begin{cases} -\Delta u = \lambda u + u^{2^*-2} v & \hbox{in} \Omega,\\ -\Delta v= \mu v^{2^*-1} + u^{2^*-1} & \hbox{in} \Omega,\\ u> 0,v> 0 & \hbox{in} \Omega,\\ u=v=0 & \hbox{on} \partial \Omega, \end{cases} \] where $\Omega$ is a bounded domain of $\mathbb{R}^N$, $N\geq 4$, $2^*=2N/(N-2)$, $\lambda\in\mathbb{R}$ and $\mu\geq 0$ and we obtain existence and nonexistence results, depending on the value of the parameters $\lambda$ and $\mu$.