Infinitely many solutions for asymptotically linear periodic Hamiltonian elliptic systems

Fukun Zhao; Leiga Zhao; Yanheng Ding

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 16, Issue: 1, page 77-91
  • ISSN: 1292-8119

Abstract

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This paper is concerned with the following periodic Hamiltonian elliptic system { - Δ ϕ + V ( x ) ϕ = G ψ ( x , ϕ , ψ ) in ℝ N , - Δ ψ + V ( x ) ψ = G ϕ ( x , ϕ , ψ ) in ℝ N , ϕ ( x ) → 0 and ψ ( x ) → 0 as | x | → ∞ . Assuming the potential V is periodic and 0 lies in a gap of σ ( - Δ + V ) , G ( x , η ) is periodic in x and asymptotically quadratic in η = ( ϕ , ψ ) , existence and multiplicity of solutions are obtained via variational approach.


How to cite

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Zhao, Fukun, Zhao, Leiga, and Ding, Yanheng. "Infinitely many solutions for asymptotically linear periodic Hamiltonian elliptic systems." ESAIM: Control, Optimisation and Calculus of Variations 16.1 (2010): 77-91. <http://eudml.org/doc/250789>.

@article{Zhao2010,
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.
},
author = {Zhao, Fukun, Zhao, Leiga, Ding, Yanheng},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Hamiltonian elliptic system; variational methods; strongly indefinite functionals; strongly indefinite functionals},
language = {eng},
month = {1},
number = {1},
pages = {77-91},
publisher = {EDP Sciences},
title = {Infinitely many solutions for asymptotically linear periodic Hamiltonian elliptic systems},
url = {http://eudml.org/doc/250789},
volume = {16},
year = {2010},
}

TY - JOUR
AU - Zhao, Fukun
AU - Zhao, Leiga
AU - Ding, Yanheng
TI - Infinitely many solutions for asymptotically linear periodic Hamiltonian elliptic systems
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/1//
PB - EDP Sciences
VL - 16
IS - 1
SP - 77
EP - 91
AB - 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.

LA - eng
KW - Hamiltonian elliptic system; variational methods; strongly indefinite functionals; strongly indefinite functionals
UR - http://eudml.org/doc/250789
ER -

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