Existence of least energy positive solutions to Schrödinger systems with mixed competition and cooperation terms: the critical case

Abstract: In this paper we investigate the existence of solutions to the following Schr\"{o}dinger system in the critical case \begin{equation*} -\Delta u_{i}+\lambda_{i}u_{i}=u_{i}\sum_{j = 1}^{d}\beta_{ij}u_{j}^{2} \text{ in } \Omega, \quad u_i=0 \text{ on } \partial \Omega, \qquad i=1,...,d. \end{equation*} Here, $\Omega\subset \mathbb{R}^{4}$ is a smooth bounded domain, $d\geq 2$, $-\lambda_{1}(\Omega)<\lambda_{i}<0$ and $\beta_{ii}>0$ for every $i$, $\beta_{ij}=\beta_{ji}$ for $i\neq j$, where $\lambda_{1}(\Omega)$ is the first eigenvalue of $-\Delta$ with Dirichlet boundary conditions. Under the assumption that the components are divided into $m$ groups, and that $\beta_{ij}\geq 0$ (cooperation) whenever components $i$ and $j$ belong to the same group, while $\beta_{ij}<0$ or $\beta_{ij}$ is positive and small (competition or weak cooperation) for components $i$ and $j$ belonging to different groups, we establish the existence of nonnegative solutions with $m$ nontrivial components, as well as classification results. Moreover, under additional assumptions on $\beta_{ij}$, we establish existence of least energy positive solutions in the case of mixed cooperation and competition. The proof is done by induction on the number of groups, and requires new estimates comparing energy levels of the system with those of appropriate sub-systems. In the case $\Omega=\mathbb{R}^4$ and $\lambda_1=\ldots=\lambda_m=0$, we present new nonexistence results. This paper can be seen as the counterpart of [Soave-Tavares, J. Differential Equations 261 (2016), 505-537] in the critical case, while extending and improving some results from [Chen-Zou, Arch. Ration. Mech. Anal. 205 (2012), 515--551], [Guo-Luo-Zou, Nonlinearity 31 (2018), 314--339].