The concentration behavior of ground states solution for a Schrödinger system of Hamiltonian type

Abstrakt

In this paper, we consider the following nonlinear Schrödinger system of Hamiltonian type \begin{equation*} \begin{cases} -\varepsilon^2 \Delta u+u+V(x)v=H_v(u,v), & x\in \mathbb{R}^N, \\ -\varepsilon^2 \Delta v+v+V(x) u= H_u(u, v),& x\in \mathbb{R}^N,\\ u(x)\rightarrow0 \text{ and } v(x)\rightarrow0 & \text{as}\ |x|\rightarrow\infty, \end{cases} \end{equation*} where $\varepsilon> 0$ is a small parameter, $V\in C^1(\mathbb{R}^N,\ \mathbb{R})$, $H\in C^1(\mathbb{R}\times\mathbb{R},\mathbb{R})$ and $(u,v)\in\mathbb{R}^2$. Under only a local condition that $V$ has a local trapping potential well, a ground state $z_\varepsilon=(u_\varepsilon,v_\varepsilon)$ of the above Schrödinger system is obtained via a combination of linking-type arguments with the generalized Nehari manifold. Moreover, we also show that ground state solution $z_\varepsilon$ concentrating around the local minimum points of the potential $V$ as $\varepsilon\rightarrow 0^+$.