This paper is concerned with the following Hamiltonian elliptic system −Δu+V(x)u=Wv(x,u,v), x∈RN,−Δv+V(x)v=Wu(x,u,v), x∈RN, $$ \left\{ \begin{array}{ll} -\Delta u+V(x)u=W_{v}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N},\\ -\Delta v+V(x)v=W_{u}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N},\\ \end{array} \right. $$ where z = ( u , v ) : ℝ N → ℝ 2 , 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):=12V∞(x)|Az|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 inf ℝ N V ∞ ( x ) > min ℝ N V ( x ), and A is a symmetric non-negative definite matrix.

中文翻译：

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哈密​​顿椭圆系统的新渐近二次条件

本文关注如下哈密顿椭圆系统-Δu+V(x)u=Wv(x,u,v), x∈RN,-Δv+V(x)v=Wu(x,u,v)， x∈RN, $$ \left\{ \begin{array}{ll} -\Delta u+V(x)u=W_{v}(x, u, v), \ \ \ \ x\in {\ mathbb{R}}^{N},\\ -\Delta v+V(x)v=W_{u}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^ {N},\\ \end{数组} \right。$$ 其中 z = ( u , v ) : ℝ N → ℝ 2 , V ( x ) 和 W ( x , z ) 在 x 中是 1 周期的。通过对强不定问题利用变分法，我们得到了非线性W(x,z):=12V∞(x)|Az|2+F(x,z)的新条件下非平凡解的新存在结果$ W(x,z):=\frac{1}{2}V_{\infty}(x)|Az|^2+F(x, z) $ 是一般渐近二次方，其中 V ∞ ( x ) ∈ (ℝ N , ℝ) 在 x 和 inf 中是 1 周期的 ℝ NV ∞ ( x ) > min ℝ NV ( x )，A 是一个对称非负定矩阵。