Existence of least energy positive solutions to Schr\"{o}dinger systems with mixed competition and cooperation terms: the critical case

TL;DR

This paper proves the existence of least energy positive solutions for a critical Schrödinger system with mixed cooperation and competition terms, extending previous results and providing new nonexistence results in certain cases.

Contribution

It establishes the existence of solutions in the critical case for systems with mixed cooperation and competition, using induction and new energy estimates.

Findings

Existence of nonnegative solutions with multiple nontrivial components.

Classification results for solutions based on component grouping.

New nonexistence results in the case of zero eigenvalues.

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].