Positive solutions for critically coupled Schrödinger systems with attractive interactions

In this paper, we consider the following coupled Schrödinger system with doubly critical exponents:

$\left\{ {\begin{array}{*{20}{l}}{ - \Delta u + {\lambda _1}u = {\mu _1}{u^3} + \beta u{v^2},}&{x \in \Omega ,}\\{ - \Delta v + {\lambda _2}v = {\mu _2}{v^3} + \beta v{u^2},}&{x \in \Omega ,}\\{u,v \ge 0,}&{x \in \Omega ,}\\{u = v = 0,}&{x \in \partial \Omega ,}\end{array}} \right.$

where $Ω\subset\mathbb R^4$ is a smooth bounded domain, $μ_1, μ_2>0$ and $β>0$, $-λ_1(Ω)<λ_1, λ_2<0$, here $λ_1(Ω)$ is the first eigenvalue of $-Δ$ with the Dirichlet boundary condition. We give the optimal ranges of $β>0$ for the existence of positive solutions to the problem, which is an open problem proposed by Chen and Zou in [Arch. Rational Mech. Anal. 205 (2012), 515-551]. Finally, as a by-product of our approaches, we extend the existence results to a critically coupled Schrödinger system defined in the whole space:

$\left\{ \begin{array}{l} - \Delta u + u = {\mu _1}{u^3} + \beta u{v^2} + f(u),\;\;\;\;x \in {\mathbb R^4},\\ - \Delta v + v = {\mu _2}{v^3} + \beta v{u^2} + g(v),\;\;\;\;\;x \in {\mathbb R^4}.\end{array} \right.$