Standing waves for a coupled system of nonlinear Schrödinger equations

We study the following system of nonlinear Schrodinger equations: $$\begin{aligned} \left\{ \begin{array}{l} -\varepsilon ^2\Delta u +a(x) u = f(u)+\lambda v, \quad x\in \mathbb R ^N, \\ -\varepsilon ^2\Delta v +b(x) v =g(v)+\lambda u, \quad x\in \mathbb R ^N,\\ u,v >0 \,\,\,\hbox {in}\,\,\,\mathbb R ^N,\quad u, v \in H^1 (\mathbb R ^N), \end{array}\right. \end{aligned}$$ where \(N\ge 3\), \(\varepsilon , \lambda >0\), and \(a, b, f, g\) are continuous functions. Under very general assumptions on both the potentials \(a, b\) and the nonlinearities \(f, g\), for small \(\lambda >0\) and \(\varepsilon >0\), we obtain positive solutions of this coupled system via pure variational methods. The asymptotic behaviors of these solutions are also studied either as \(\varepsilon \rightarrow 0\) or as \(\lambda \rightarrow 0\).