Classification of Ground-states of a coupled Schrödinger system

The paper is concerned with the existence of nontrivial ground-state solutions for a coupled nonlinear Schrodinger system $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u_j+ \lambda u_j=\mu |u_j|^{2p}u_j+\sum _{i\ne j}^m \beta |u_i|^{p+1}|u_j|^{p-1}u_j, &{} \text {in}\ \mathbb {R}^n, \\ u_j(x)\rightarrow 0\ \text {as}\ |x|\ \rightarrow \infty , \quad j=1,2,\ldots , m, \end{array}\right. \end{aligned}$$ where $$ m\ge 2$$ , $$00,$$ $$\mu >0$$ and $$\beta >0$$ . We establish a sufficient and necessary condition for the existence of nontrivial ground-state solutions which have the least energy among all the non-zero solutions of the system and whose components have the same modulus. This gives an affirmative answer to a conjecture raised in Correia (Nonlinear Anal. 140:112–129, 2016).