New asymptotically quadratic conditions for Hamiltonian elliptic systems

This paper is concerned with the following Hamiltonian elliptic system − Δ u + V ( x ) u = W v ( x , u , v ) , x ∈ R N , − Δ v + V ( x ) v = W u ( x , u , v ) , x ∈ R N , $$ \left\{ \begin{array}{ll} -\Delta u+V(x)u=W_{v}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N},\\ -\Delta v+V(x)v=W_{u}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N},\\ \end{array} \right. $$ where z = (u, v) : ℝ N → ℝ2, V(x) and W(x, z) are 1-periodic in x. By making use of variational approach for strongly indefinite problems, we obtain a new existence result of nontrivial solution under new conditions that the nonlinearity W ( x , z ) := 1 2 V ∞ ( x ) | A z | 2 + F ( x , z ) $ W(x,z):=\frac{1}{2}V_{\infty}(x)|Az|^2+F(x, z) $ is general asymptotically quadratic, where V ∞(x) ∈ (ℝ N , ℝ) is 1-periodic in x and infℝ N V ∞(x) > minℝ N V(x), and A is a symmetric non-negative definite matrix.