Existence and decay property of ground state solutions for Hamiltonian elliptic system

Abstract

In this paper we study the following nonlinear Hamiltonian elliptic system with gradient term \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{ll} -\Delta u +\vec{b}(x)\cdot \nabla u +u+V(x)v = f(x, |z|)v, \; \; x\in\mathbb{R}^{N}, -\Delta v -\vec{b}(x)\cdot \nabla v +v+V(x)u = f(x, |z|)u, \; \; x\in\mathbb{R}^{N}, \end{array} \right. \end{eqnarray*} $\end{document} where \begin{document}$ z = (u, v)\in\mathbb{R}^{2} $\end{document} . Under some suitable conditions on the potential and nonlinearity, we obtain the existence of ground state solutions in periodic case and asymptotically periodic case via variational methods, respectively. Moreover, we also explore some properties of these ground state solutions, such as compactness of set of ground state solutions and exponential decay of ground state solutions.