Periodic Hamiltonean Elliptic Systems in Unbounded Domains: Superquadratic and Asymptotically Quadratic

We study the system of elliptic equations\[\left\{\begin{array}{rcl}-\Delta u + b(x)\cdot \nabla u + V (x)u &=& H_v(x; u; v) \\-\Delta v + b(x)\cdot \nabla v + V (x)v &=& H_u(x; u; v)\\\end{array}\right. \qquad \mbox{\textrm{for $x \in \mathbb{R}^N$}}\]where $z = (u; v) \colon \mathbb{R}^N \to \mathbb{R}^m \times \mathbb{R}^m$, $b\in C^1(\mathbb{R}^N;\mathbb{R}^N)$ with the gauge condition $div b = 0$, $V\in C(\mathbb{R}^N;\mathbb{R})$ and $H\in C^1(\mathbb{R}^N \times \mathbb{R}^{2m};R)$. Assume the system depends periodically on the variable $x$ and $H(x; z)$ is asymptotically quadratic or super quadratic with nonlinear conditions as $|z|\to\infty$ more general than the Ambrosetti-Rabinowitz condition.We establish the existence of solutions in $H^1(\mathbb{R}^N;\mathbb{R}^{2m})$. If moreover $H(x; z)$ is even in $z$ we obtain infinitely many such solutions.