New existence and symmetry results for least energy positive solutions of Schrödinger systems with mixed competition and cooperation terms

Abstract: In this paper we focus on existence and symmetry properties of solutions to the cubic Schr\"odinger system [ -\Delta u_i +\lambda_i u_i = \sum_{j=1}^d \beta_{ij} u_j^2 u_i \quad \text{in $\Omega \subset \mathbb{R}^N$},\qquad i=1,\dots d ] where $d\geq 2$, $\lambda_i,\beta_{ii}>0$, $\beta_{ij}=\beta_{ji}\in \mathbb{R}$ for $j\neq i$, $N=2,3$. The underlying domain $\Omega$ is either bounded or the whole space, and $u_i\in H^1_0(\Omega)$ or $u_i\in H^1_{rad}(\mathbb{R}^N)$ respectively. We establish new existence and symmetry results for least energy positive solutions in the case of mixed cooperation and competition coefficients, as well as in the purely cooperative case.