ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - January 2010 - Volume 16, Issue 01  

Research Article

Infinitely many solutions for asymptotically linear periodic Hamiltonian elliptic systems

Zhao, Fukuna1a2, Zhao, Leigaa3 and Ding, Yanhenga2

a1 Department of Mathematics, Yunnan Normal University, Kunming 650092 Yunnan, P.R. China. fukunzhao@163.com

a2 Institute of Mathematics, AMSS, CAS, Beijing 100080, P.R. China.

a3 Department of Mathematics, Beijing University of Chemical technology, Beijing 100029, P.R. China.

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.


(Received March 11 2008)

(Revised July 6 2008)

(Online publication October 21 2008)

Key Words:

  • Hamiltonian elliptic system;
  • variational methods;
  • strongly indefinite functionals

Mathematics Subject Classification:

  • 35J50;
  • 35J55