Synchronized and ground-state solutions to a coupled Schrödinger system

<p style='text-indent:20px;'>In this paper, we study the following coupled nonlinear Schrödinger system of the form</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \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. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>for <inline-formula><tex-math id="M1">\begin{document}$ m = 2,3 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ \Omega\subset \mathbb{R}^N $\end{document}</tex-math></inline-formula> is a bounded domain or <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ N\geq 3 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ F(t_1,t_2\cdots,t_m)\in C^1(\mathbb{R}^m,\mathbb{R}) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ \kappa_i\in\mathbb{R} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ g_i\in C(\mathbb{R}) \ (i = 1,2,\cdots,m) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula> 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.</p>