Synchronized and ground-state solutions to a coupled Schrödinger system

In this paper, we study the following coupled nonlinear Schrödinger system of the form\begin{document}$ \left\{\begin{array}{l} -\Delta u_i-\kappa_iu_i = g_i(u_i)+\lambda\partial_iF(\vec{u}), \\ \vec{u} = (u_1,u_2,\cdots,u_m), u_i\in D_0^{1,2}(\Omega), \end{array}\right. $\end{document}for \begin{document}$ m = 2,3 $\end{document} , where \begin{document}$ \Omega\subset \mathbb{R}^N $\end{document} is a bounded domain or \begin{document}$ \mathbb{R}^N $\end{document} , \begin{document}$ N\geq 3 $\end{document} , \begin{document}$ F(t_1,t_2\cdots,t_m)\in C^1(\mathbb{R}^m,\mathbb{R}) $\end{document} , \begin{document}$ \kappa_i\in\mathbb{R} $\end{document} , \begin{document}$ g_i\in C(\mathbb{R}) \ (i = 1,2,\cdots,m) $\end{document} and \begin{document}$ \lambda>0 $\end{document} is large enough. In this work we mainly focus on the existence of fully nontrivial ground-state solutions and synchronized ground-state solutions under certain conditions.