# 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

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topZhao, 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 -

## References

top- N. Ackermann, On a periodic Schrödinger equation with nonlinear superlinear part. Math. Z.248 (2004) 423–443. Zbl1059.35037
- N. Ackermann, A superposition principle and multibump solutions of periodic Schrödinger equations. J. Func. Anal.234 (2006) 277–320. Zbl1126.35057
- C.O. Alves, P.C. Carrião and O.H. Miyagaki, On the existence of positive solutions of a perturbed Hamiltonian system in ${\mathbb{R}}^{N}$. J. Math. Anal. Appl.276 (2002) 673–690. Zbl1056.35060
- A.I. Ávila and J. Yang, On the existence and shape of least energy solutions for some elliptic systems. J. Diff. Eq.191 (2003) 348–376. Zbl1109.35325
- A.I. Ávila and J. Yang, Multiple solutions of nonlinear elliptic systems. Nonlinear Differ. Equ. Appl.12 (2005) 459–479. Zbl1146.35346
- T. Bartsch and D.G. De Figueiredo, Infinitely many solutions of nonlinear elliptic systems, in Progress in Nonlinear Differential Equations and Their Applications35, Birkhäuser, Basel/Switzerland (1999) 51–67. Zbl0922.35049
- T. Bartsch and Y. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory. Math. Nach.279 (2006) 1–22. Zbl1117.58007
- V. Benci and P.H. Rabinowitz, Critical point theorems for indefinite functionals. Inven. Math.52 (1979) 241–273. Zbl0465.49006
- V. Coti-Zelati and P.H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials. J. Amer. Math. Soc.4 (1991) 693–727. Zbl0744.34045
- V. Coti-Zelati and P.H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on ${\mathbb{R}}^{N}$. Comm. Pure Appl. Math.45 (1992) 1217–1269. Zbl0785.35029
- D.G. De Figueiredo and Y.H. Ding, Strongly indefinite functionals and multiple solutions of elliptic systems. Trans. Amer. Math. Soc.355 (2003) 2973–2989. Zbl1125.35338
- D.G. De Figueiredo and P.L. Felmer, On superquadratic elliptic systems. Trans. Amer. Math. Soc.343 (1994) 97–116. Zbl0799.35063
- D.G. De Figueiredo and J. Yang, Decay, symmetry and existence of solutions of semilinear elliptic systems. Nonlinear Anal.33 (1998) 211–234. Zbl0938.35054
- D.G. De Figueiredo, J. Marcos do Ó and B. Ruf, An Orlicz-space approach to superlinear elliptic systems. J. Func. Anal.224 (2005) 471–496. Zbl1210.35081
- Y. Ding and L. Jeanjean, Homoclinic orbits for a non periodic Hamiltonian system. J. Diff. Eq.237 (2007) 473–490. Zbl1117.37032
- Y. Ding and F.H. Lin, Semiclassical states of Hamiltonian systems of Schrödinger equations with subcritical and critical nonlinearies. J. Partial Diff. Eqs.19 (2006) 232–255. Zbl1104.35051
- J. Hulshof and R.C.A.M. Van de Vorst, Differential systems with strongly variational structure. J. Func. Anal.114 (1993) 32–58. Zbl0793.35038
- W. Kryszewski and A. Szulkin, An infinite dimensional Morse theory with applications. Trans. Amer. Math. Soc.349 (1997) 3181–3234. Zbl0892.58015
- W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equations. Adv. Differential Equations3 (1998) 441–472. Zbl0947.35061
- G. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part. Comm. Contemp. Math.4 (2002) 763–776. Zbl1056.35065
- G. Li and J. Yang, Asymptotically linear elliptic systems. Comm. Partial Diff. Eq.29 (2004) 925–954. Zbl1140.35406
- A. Pistoia and M. Ramos, Locating the peaks of the least energy solutions to an elliptic system with Neumann boundary conditions. J. Diff. Eq.201 (2004) 160–176. Zbl1246.35089
- M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV Analysis of Operators. Academic Press, New York (1978). Zbl0401.47001
- E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian stysems. Math. Z.209 (1992) 133–160.
- B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in RN. Adv. Differential Equations5 (2000) 1445–1464. Zbl1213.35223
- C. Troestler and M. Willem, Nontrivial solution of a semilinear Schrödinger equation. Comm. Partial Diff. Eq.21 (1996) 1431–1449. Zbl0864.35036
- M. Willem, Minimax Theorems. Birkhäuser, Berlin (1996).
- J. Yang, Nontrivial solutions of semilinear elliptic systems in ${\mathbb{R}}^{N}$. Electron. J. Diff. Eqns.6 (2001) 343–357. Zbl1099.35514

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