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

Abstract

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

19 pages, pre-peer version, to appear in Calc. Var. Partial Differential Equations