This paper is concerned with the following non-periodic diffusion
system
$\partial_tu-\Delta_x u+b(t,x)\cdot\nabla_x u+V(x)u=H_v(t,x,u,v)$in $\mathbb{R}\times\mathbb{R}^N,$
$-\partial_tv-\Delta_x v-b(t,x)\cdot\nabla_x v+V(x)v=H_u(t,x,u,v)$in$\mathbb{R}\times\mathbb{R}^N,$
$u(t,x)\to 0$and$v(t,x)\to0$as$|t|+|x|\to\infty.$
Assuming the potential $V$ is bounded and has a positive bound from
below, existence and multiplicity of solutions are obtained for the
system with asymptotically quadratic
nonlinearities via variational approach.
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