Ground State Solutions of Nehari-Pankov Type for a Superlinear Hamiltonian Elliptic System on ${mathbb{R}}^{N}$

摘要

This paper is concerned with the followingelliptic system of Hamiltonian type[ left{ egin{array}{ll} - riangle u+V(x)u=W_{v}(x, u, v), xin {mathbb{R}}^{N}, - riangle v+V(x)v=W_{u}(x, u, v), xin {mathbb{R}}^{N}, u, vin H^{1}({mathbb{R}}^{N}), end{array} ight. ] where the potential $V$ is periodic and $0$ lies in a gap ofthe spectrum of $-Delta+V$, $W(x, s, t)$ is periodic in $x$ and superlinear in $s$ and $t$ at infinity.We develop a direct approach to find ground state solutions of Nehari-Pankov type for the above system.Especially, our method is applicable for the case when [ W(x, u, v)=sum_{i=1}^{k}int_{0}^{|alpha_iu+eta_iv|}g_i(x,t)tmathrm{d}t +sum_{j=1}^{l}int_{0}^{sqrt{u^2+2b_juv+a_jv^2}}h_j(x,t)tmathrm{d}t, ] where $alpha_i, eta_i, a_j, b_jin mathbb{R}$ with $alpha_i^2+eta_i^2 e0$ and $a_jgt b_j^2$, $g_i(x, t)$ and $h_j(x, t)$ are nondecreasing in $tin mathbb{R}^{+}$ for every$xin mathbb{R}^N$ and $g_i(x, 0)=h_j(x, 0)=0$.