New conditions on ground state solutions for Hamiltonian elliptic systems with gradient terms

‎This paper is concerned with the following elliptic system‎: ‎$$‎ ‎\left\{‎ ‎\begin{array}{ll}‎ -‎\triangle u‎ + ‎b(x)\nabla u‎ + ‎V(x)u=g(x‎, ‎v)‎, ‎\\‎ -‎\triangle v‎ - ‎b(x)\nabla v‎ + ‎V(x)v=f(x‎, ‎u)‎, ‎\\‎ ‎\end{array}‎ ‎\right‎. ‎$$‎ ‎for $x \in {\mathbb{R}}^{N}$‎, ‎where $V $‎, ‎$b$ and $W$ are 1-periodic in $x$‎, ‎and $f(x,t)$‎, ‎$g(x,t)$ are Superlinear‎. ‎In this paper‎, ‎we give a new technique to show the boundedness of Cerami sequences and establish the existence of ground state solutions with mild assumptions on $f$ and $g$.