Ground States for a nonlinear Schr\"odinger system with sublinear coupling terms

Abstract

We study the existence of ground states for the coupled Schr\"odinger system \begin{equation} \left\{\begin{array}{lll} \displaystyle -\Delta u_i+\lambda_i u_i= \mu_i |u_i|^{2q-2}u_i+\sum_{j\neq i}b_{ij} |u_j|^q|u_i|^{q-2}u_i \\ u_i\in H^1(\mathbb{R}^n), \quad i=1,\ldots, d, \end{array}\right. \end{equation} $n\geq 1$, for $\lambda_i,\mu_i >0$, $b_{ij}=b_{ji}>0$ (the so-called "symmetric attractive case") and $1<q<n/(n-2)^+$. We prove the existence of a nonnegative ground state $(u_1^*,\ldots,u_d^*)$ with $u_i^*$ radially decreasing. Moreover we show that, for $1<q<2$, such ground states are positive in all dimensions and for all values of the parameters.