Multiple solutions for superlinear elliptic systems of Hamiltonian type

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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

Abstract
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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
[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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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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