Infinitely many solutions for asymptotically linear periodic hamiltonian elliptic systems

Zhao, Fukun; Zhao, Leiga; Ding, Yanheng

ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 1, pp. 77-91.

This paper is concerned with the following periodic hamiltonian elliptic system $\{-\Delta \phi +V\left(x\right)\phi =G_{\psi }(x,\phi ,\psi )\text{in}{ℝ}^N,-\Delta \psi +V\left(x\right)\psi =G_{\phi }(x,\phi ,\psi )\text{in}{ℝ}^N,\phi \left(x\right)\to 0\text{and}\psi \left(x\right)\to 0\text{as}|x|\to \infty .$ Assuming the potential V is periodic and 0 lies in a gap of $\sigma (-\Delta +V)$, $G(x,\eta )$ is periodic in x and asymptotically quadratic in $\eta =(\phi ,\psi )$, existence and multiplicity of solutions are obtained via variational approach.

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