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Infinitely many solutions for asymptotically linear periodicHamiltonian elliptic systems

Published online by Cambridge University Press:  21 October 2008

Fukun Zhao ,
Leiga Zhao  and
Yanheng Ding
Fukun Zhao
Affiliation:
Department of Mathematics, Yunnan Normal University, Kunming 650092 Yunnan, P.R. China. fukunzhao@163.com Institute of Mathematics, AMSS, CAS, Beijing 100080, P.R. China.
Leiga Zhao
Affiliation:
Department of Mathematics, Beijing University of Chemical technology, Beijing 100029, P.R. China.
Yanheng Ding
Affiliation:
Institute of Mathematics, AMSS, CAS, Beijing 100080, P.R. China.
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Abstract

This paper is concerned with the following periodic Hamiltonian elliptic system

$ \{  -\Delta \varphi+V(x)\varphi=G_\psi(x,\varphi,\psi)\ \hbox{in }\mathbb{R}^N, \\ -\Delta \psi+V(x)\psi=G_\varphi(x,\varphi,\psi)\ \hbox{in }\mathbb{R}^N, \\ \varphi(x)\to 0\ \hbox{and }\psi(x)\to0\ \hbox{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 $\eta=(\varphi,\psi)$, existence and multiplicity of solutions are obtained via variational approach.


Keywords

Hamiltonian elliptic systemvariational methodsstrongly indefinite functionals
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 16 , Issue 1 , January 2010 , pp. 77 - 91
Copyright
© EDP Sciences, SMAI, 2008

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