New existence and symmetry results for least energy positive solutions of Schr\"odinger systems with mixed competition and cooperation terms

In this paper we focus on existence and symmetry properties of solutions to the cubic Schr"odinger system $$-\mathrm{\Delta }{u}_i+{\lambda }_i{u}_i=\sum _{j=1}^d{\beta }_{ij}{u}_j^2{u}_i\text{in }\mathrm{\Omega }\subset {\mathbb{R}}^N,i=1,\dots d$$ where $d\ge 2$, ${\lambda }_i,{\beta }_{ii}>0$, ${\beta }_{ij}={\beta }_{ji}\in \mathbb{R}$ for $j\mathrm{\ne }i$, $N=2,3$. The underlying domain $\mathrm{\Omega }$ 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.