Positive ground states for a system of Schrödinger equations with critically growing nonlinearities

Abstract

We study the following problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u = \lambda u + u^{2^*-2} v &{}\quad \hbox {in }\quad \Omega ,\\ -\Delta v= \mu v^{2^*-1} + u^{2^*-1} &{}\quad \hbox {in }\quad \Omega ,\\ u> 0,v> 0 &{}\quad \hbox {in }\quad \Omega ,\\ u=v=0 &{}\quad \hbox {on }\quad \partial \Omega , \end{array}\right. } \end{aligned}$$

where $\Omega$ is a bounded domain of $\mathbb {R}^N$, $N\ge 4$, $2^*=2N/(N-2)$, $\lambda \in \mathbb {R}$ and $\mu \ge 0$ and we obtain existence and nonexistence results, depending on the value of the parameters $\lambda$ and $\mu$.