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November  2011, 30(4): 1249-1262. doi: 10.3934/dcds.2011.30.1249

Multiple solutions for superlinear elliptic systems of Hamiltonian type

Rumei Zhang 1, , Jin Chen 2, and  Fukun Zhao 1,

1. 

Department of Mathematics, Yunnan Normal University, Kunming 650092 Yunnan
2. 

Department of Mathematics, Zhaotong Teacher’s College, Zhaotong 657000 Yunnan

Received  March 2010 Revised  May 2010 Published  May 2011

This paper is concerned with the following periodic Hamiltonian elliptic system

$\-\Delta \varphi+V(x)\varphi=G_\psi(x,\varphi,\psi)$ in $\mathbb{R}^N,$
$\-\Delta \psi+V(x)\psi=G_\varphi(x,\varphi,\psi)$ in $\mathbb{R}^N,$
$\varphi(x)\to 0$ and $\psi(x)\to0$ 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 superquadratic in $\eta=(\varphi,\psi)$, existence and multiplicity of solutions are obtained via variational approach.
Keywords: strongly indefinite functionals., Hamiltonian elliptic system, variational methods.
Mathematics Subject Classification: Primary: 35J50; Secondary: 35J5.
Citation: Rumei Zhang, Jin Chen, Fukun Zhao. Multiple solutions for superlinear elliptic systems of Hamiltonian type. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1249-1262. doi: 10.3934/dcds.2011.30.1249
References:
[1]

N. Ackermann, On a periodic Schrödinger equation with nonlinear superlinear part,, Math. Z., 248 (2004), 423.  doi: 10.1007/s00209-004-0663-y.  Google Scholar

[2]

N. Ackermann, A superposition principle and multibump solutions of periodic Schrödinger equations,, J. Func. Anal., 234 (2006), 277.  doi: 10.1016/j.jfa.2005.11.010.  Google Scholar

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C. O. Alves, P. C. Carrião and O. H. Miyagaki, On the existence of positive solutions of a perturbed Hamiltonian system in $\mathbbR$N,, J. Math. Anal. Appl., 276 (2002), 673.  doi: 10.1016/S0022-247X(02)00413-4.  Google Scholar

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A. I. Ávila and J. Yang, Multiple solutions of nonlinear elliptic systems,, Nonlinear Differ. Equ. Appl., 12 (2005), 459.   Google Scholar

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A. I. Ávila and J. Yang, On the existence and shape of least energy solutions for some elliptic systems,, J. Differential Equations, 191 (2003), 348.   Google Scholar

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T. Bartsch and D. G. De Figueiredo, Infinitely many solutions of nonlinear elliptic systems,, in, 35 (1999), 51.   Google Scholar

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T. Bartsch and Y. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory,, Math. Nach., 279 (2006), 1.  doi: 10.1002/mana.200410420.  Google Scholar

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V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals,, Inven. Math., 52 (1979), 241.  doi: 10.1007/BF01389883.  Google Scholar

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V. Coti-Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials,, J. Amer. Math. Soc., 4 (1991), 693.  doi: 10.1090/S0894-0347-1991-1119200-3.  Google Scholar

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V. Coti-Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $\mathbbR$N,, Comm. Pure Appl. Math., 45 (1992), 1217.  doi: 10.1002/cpa.3160451002.  Google Scholar

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D. G. De Figueiredo and Y. H. Ding, Strongly indefinite functionals and multiple solutions of elliptic systems,, Tran. Amer. Math. Soc., 355 (2003), 2973.  doi: 10.1090/S0002-9947-03-03257-4.  Google Scholar

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D. G. De Figueiredo and P. L. Felmer, On superquadratic elliptic systems,, Tran. Amer. Math. Soc., 343 (1994), 97.   Google Scholar

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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.  doi: 10.1016/j.jfa.2004.09.008.  Google Scholar

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D. G. De Figueiredo and J. Yang, Decay, symmetry and existence of solutions of semilinear elliptic systems,, Nonlinear Anal., 33 (1998), 211.  doi: 10.1016/S0362-546X(97)00548-8.  Google Scholar

[15]

Y. Ding, "Variational Methods for Strongly Indefinite Problems,", Interdisciplinary Mathematical Sciences, 7 (2007).  doi: 10.1142/9789812709639.  Google Scholar

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Y. Ding and L. Jeanjean, Homoclinic orbits for a non periodic Hamiltonian system,, J. Differential Equations, 237 (2007), 473.  doi: 10.1016/j.jde.2007.03.005.  Google Scholar

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Y. Ding and C. Lee, Existence and exponential decay of homoclinics in a nonperiodic superquadratic Hamiltonian system,, J. Differential Equations, 246 (2009), 2829.   Google Scholar

[18]

J. Hulshof and R. C. A. M. Van de Vorst, Differential systems with strongly variational structure,, J. Func. Anal., 114 (1993), 32.  doi: 10.1006/jfan.1993.1062.  Google Scholar

[19]

W. Kryszewski and A. Szulkin, An infinite dimensional Morse theory with applications,, Tran. Amer. Math. Soc., 349 (1997), 3181.  doi: 10.1090/S0002-9947-97-01963-6.  Google Scholar

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W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equations,, Adv. Differential Equations, 3 (1998), 441.   Google Scholar

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G. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part,, Comm. Contemp. Math., 4 (2002), 763.  doi: 10.1142/S0219199702000853.  Google Scholar

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G. Li and J. Yang, Asymptotically linear elliptic systems,, Comm. Partial Differential Equations, 29 (2004), 925.   Google Scholar

[23]

A. Pistoia and M. Ramos, Locating the peaks of the least energy solutions to an ellyptic system with Neumann boundary conditions,, J. Differential Equations, 201 (2004), 160.   Google Scholar

[24]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics, IV Analysis of Operators,", Academic Press, (1978).   Google Scholar

[25]

E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian stysems,, Math. Z., 209 (1992), 133.   Google Scholar

[26]

B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $R$N,, Adv. Differential Equations, 5 (2000), 1445.   Google Scholar

[27]

C. Troestler and M. Willem, Nontrivial solution of a semilinear Schrödinger equation,, Comm. Partial Differential Equations, 21 (1996), 1431.   Google Scholar

[28]

J. Wang, J. Xu and F. Zhang, Existence of solutions for nonperiodic superquadratic Hamiltonian elliptic systems,, Nonlinear Anal., 72 (2010), 1949.  doi: 10.1016/j.na.2009.09.035.  Google Scholar

[29]

M. Willem, "Minimax Theorems,", Birkhäuser, (1996).   Google Scholar

[30]

J. Yang, Nontrivial solutions of semilinear elliptic systems in $\mathbbR$N,, Electron. J. Diff. Eqns., 6 (2001), 343.   Google Scholar

[31]

F. Zhao, L. Zhao and Y. Ding, Multiple solutions for asymptotically linear elliptic systems,, Nonlinear Differ. Equ. Appl., 15 (2008), 673.   Google Scholar

[32]

F. Zhao, L. Zhao and Y. Ding, Infinitely many solutions for asymptotically linear periodic Hamiltonian ellitpic systems,, ESAIM: Control, 16 (2010), 77.  doi: 10.1051/cocv:2008064.  Google Scholar

show all references

References:
[1]

N. Ackermann, On a periodic Schrödinger equation with nonlinear superlinear part,, Math. Z., 248 (2004), 423.  doi: 10.1007/s00209-004-0663-y.  Google Scholar

[2]

N. Ackermann, A superposition principle and multibump solutions of periodic Schrödinger equations,, J. Func. Anal., 234 (2006), 277.  doi: 10.1016/j.jfa.2005.11.010.  Google Scholar

[3]

C. O. Alves, P. C. Carrião and O. H. Miyagaki, On the existence of positive solutions of a perturbed Hamiltonian system in $\mathbbR$N,, J. Math. Anal. Appl., 276 (2002), 673.  doi: 10.1016/S0022-247X(02)00413-4.  Google Scholar

[4]

A. I. Ávila and J. Yang, Multiple solutions of nonlinear elliptic systems,, Nonlinear Differ. Equ. Appl., 12 (2005), 459.   Google Scholar

[5]

A. I. Ávila and J. Yang, On the existence and shape of least energy solutions for some elliptic systems,, J. Differential Equations, 191 (2003), 348.   Google Scholar

[6]

T. Bartsch and D. G. De Figueiredo, Infinitely many solutions of nonlinear elliptic systems,, in, 35 (1999), 51.   Google Scholar

[7]

T. Bartsch and Y. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory,, Math. Nach., 279 (2006), 1.  doi: 10.1002/mana.200410420.  Google Scholar

[8]

V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals,, Inven. Math., 52 (1979), 241.  doi: 10.1007/BF01389883.  Google Scholar

[9]

V. Coti-Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials,, J. Amer. Math. Soc., 4 (1991), 693.  doi: 10.1090/S0894-0347-1991-1119200-3.  Google Scholar

[10]

V. Coti-Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $\mathbbR$N,, Comm. Pure Appl. Math., 45 (1992), 1217.  doi: 10.1002/cpa.3160451002.  Google Scholar

[11]

D. G. De Figueiredo and Y. H. Ding, Strongly indefinite functionals and multiple solutions of elliptic systems,, Tran. Amer. Math. Soc., 355 (2003), 2973.  doi: 10.1090/S0002-9947-03-03257-4.  Google Scholar

[12]

D. G. De Figueiredo and P. L. Felmer, On superquadratic elliptic systems,, Tran. Amer. Math. Soc., 343 (1994), 97.   Google Scholar

[13]

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.  doi: 10.1016/j.jfa.2004.09.008.  Google Scholar

[14]

D. G. De Figueiredo and J. Yang, Decay, symmetry and existence of solutions of semilinear elliptic systems,, Nonlinear Anal., 33 (1998), 211.  doi: 10.1016/S0362-546X(97)00548-8.  Google Scholar

[15]

Y. Ding, "Variational Methods for Strongly Indefinite Problems,", Interdisciplinary Mathematical Sciences, 7 (2007).  doi: 10.1142/9789812709639.  Google Scholar

[16]

Y. Ding and L. Jeanjean, Homoclinic orbits for a non periodic Hamiltonian system,, J. Differential Equations, 237 (2007), 473.  doi: 10.1016/j.jde.2007.03.005.  Google Scholar

[17]

Y. Ding and C. Lee, Existence and exponential decay of homoclinics in a nonperiodic superquadratic Hamiltonian system,, J. Differential Equations, 246 (2009), 2829.   Google Scholar

[18]

J. Hulshof and R. C. A. M. Van de Vorst, Differential systems with strongly variational structure,, J. Func. Anal., 114 (1993), 32.  doi: 10.1006/jfan.1993.1062.  Google Scholar

[19]

W. Kryszewski and A. Szulkin, An infinite dimensional Morse theory with applications,, Tran. Amer. Math. Soc., 349 (1997), 3181.  doi: 10.1090/S0002-9947-97-01963-6.  Google Scholar

[20]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equations,, Adv. Differential Equations, 3 (1998), 441.   Google Scholar

[21]

G. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part,, Comm. Contemp. Math., 4 (2002), 763.  doi: 10.1142/S0219199702000853.  Google Scholar

[22]

G. Li and J. Yang, Asymptotically linear elliptic systems,, Comm. Partial Differential Equations, 29 (2004), 925.   Google Scholar

[23]

A. Pistoia and M. Ramos, Locating the peaks of the least energy solutions to an ellyptic system with Neumann boundary conditions,, J. Differential Equations, 201 (2004), 160.   Google Scholar

[24]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics, IV Analysis of Operators,", Academic Press, (1978).   Google Scholar

[25]

E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian stysems,, Math. Z., 209 (1992), 133.   Google Scholar

[26]

B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $R$N,, Adv. Differential Equations, 5 (2000), 1445.   Google Scholar

[27]

C. Troestler and M. Willem, Nontrivial solution of a semilinear Schrödinger equation,, Comm. Partial Differential Equations, 21 (1996), 1431.   Google Scholar

[28]

J. Wang, J. Xu and F. Zhang, Existence of solutions for nonperiodic superquadratic Hamiltonian elliptic systems,, Nonlinear Anal., 72 (2010), 1949.  doi: 10.1016/j.na.2009.09.035.  Google Scholar

[29]

M. Willem, "Minimax Theorems,", Birkhäuser, (1996).   Google Scholar

[30]

J. Yang, Nontrivial solutions of semilinear elliptic systems in $\mathbbR$N,, Electron. J. Diff. Eqns., 6 (2001), 343.   Google Scholar

[31]

F. Zhao, L. Zhao and Y. Ding, Multiple solutions for asymptotically linear elliptic systems,, Nonlinear Differ. Equ. Appl., 15 (2008), 673.   Google Scholar

[32]

F. Zhao, L. Zhao and Y. Ding, Infinitely many solutions for asymptotically linear periodic Hamiltonian ellitpic systems,, ESAIM: Control, 16 (2010), 77.  doi: 10.1051/cocv:2008064.  Google Scholar

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