DCDS
Multiple solutions for superlinear elliptic systems of Hamiltonian type
Rumei Zhang Jin Chen Fukun Zhao
Discrete & Continuous Dynamical Systems - A 2011, 30(4): 1249-1262 doi: 10.3934/dcds.2011.30.1249
This paper is concerned with the following periodic Hamiltonian elliptic system

$\-\Delta \varphi+V(x)\varphi=G_\psi(x,\varphi,\psi)$ in $\mathbb{R}^N,$
$\-\Delta \psi+V(x)\psi=G_\varphi(x,\varphi,\psi)$ in $\mathbb{R}^N,$
$\varphi(x)\to 0$ and $\psi(x)\to0$ as $|x|\to\infty.$

Assuming the potential $V$ is periodic and $0$ lies in a gap of $\sigma(-\Delta+V)$, $G(x,\eta)$ is periodic in $x$ and superquadratic in $\eta=(\varphi,\psi)$, existence and multiplicity of solutions are obtained via variational approach.
keywords: strongly indefinite functionals. Hamiltonian elliptic system variational methods

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