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

Multiple solutions for superlinear elliptic systems of Hamiltonian type

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.
Citation: Rumei Zhang, Jin Chen, Fukun Zhao. Multiple solutions for superlinear elliptic systems of Hamiltonian type. Discrete and Continuous Dynamical Systems, 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), 423-443. doi: 10.1007/s00209-004-0663-y.

[2]

N. Ackermann, A superposition principle and multibump solutions of periodic Schrödinger equations, J. Func. Anal., 234 (2006), 277-320. doi: 10.1016/j.jfa.2005.11.010.

[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), 673-690. doi: 10.1016/S0022-247X(02)00413-4.

[4]

A. I. Ávila and J. Yang, Multiple solutions of nonlinear elliptic systems, Nonlinear Differ. Equ. Appl., 12 (2005), 459-479.

[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), 348-376.

[6]

T. Bartsch and D. G. De Figueiredo, Infinitely many solutions of nonlinear elliptic systems, in "Progress in Nonlinear Differential Equations and Their Applications," 35, Birkhäuser, Basel/Switzerland, (1999), 51-67.

[7]

T. Bartsch and Y. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Math. Nach., 279 (2006), 1-22. doi: 10.1002/mana.200410420.

[8]

V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals, Inven. Math., 52 (1979), 241-273. doi: 10.1007/BF01389883.

[9]

V. Coti-Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727. doi: 10.1090/S0894-0347-1991-1119200-3.

[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), 1217-1269. doi: 10.1002/cpa.3160451002.

[11]

D. G. De Figueiredo and Y. H. Ding, Strongly indefinite functionals and multiple solutions of elliptic systems, Tran. Amer. Math. Soc., 355 (2003), 2973-2989. doi: 10.1090/S0002-9947-03-03257-4.

[12]

D. G. De Figueiredo and P. L. Felmer, On superquadratic elliptic systems, Tran. Amer. Math. Soc., 343 (1994), 97-116.

[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), 471-496. doi: 10.1016/j.jfa.2004.09.008.

[14]

D. G. De Figueiredo and J. Yang, Decay, symmetry and existence of solutions of semilinear elliptic systems, Nonlinear Anal., 33 (1998), 211-234. doi: 10.1016/S0362-546X(97)00548-8.

[15]

Y. Ding, "Variational Methods for Strongly Indefinite Problems," Interdisciplinary Mathematical Sciences, 7, World Sci., 2007. doi: 10.1142/9789812709639.

[16]

Y. Ding and L. Jeanjean, Homoclinic orbits for a non periodic Hamiltonian system, J. Differential Equations, 237 (2007), 473-490. doi: 10.1016/j.jde.2007.03.005.

[17]

Y. Ding and C. Lee, Existence and exponential decay of homoclinics in a nonperiodic superquadratic Hamiltonian system, J. Differential Equations, 246 (2009), 2829-2848.

[18]

J. Hulshof and R. C. A. M. Van de Vorst, Differential systems with strongly variational structure, J. Func. Anal., 114 (1993), 32-58. doi: 10.1006/jfan.1993.1062.

[19]

W. Kryszewski and A. Szulkin, An infinite dimensional Morse theory with applications, Tran. Amer. Math. Soc., 349 (1997), 3181-3234. doi: 10.1090/S0002-9947-97-01963-6.

[20]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equations, Adv. Differential Equations, 3 (1998), 441-472.

[21]

G. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Comm. Contemp. Math., 4 (2002), 763-776. doi: 10.1142/S0219199702000853.

[22]

G. Li and J. Yang, Asymptotically linear elliptic systems, Comm. Partial Differential Equations, 29 (2004), 925-954.

[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), 160-176.

[24]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics, IV Analysis of Operators," Academic Press, New York, 1978.

[25]

E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian stysems, Math. Z., 209 (1992), 133-160.

[26]

B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $R$N, Adv. Differential Equations, 5 (2000), 1445-1464.

[27]

C. Troestler and M. Willem, Nontrivial solution of a semilinear Schrödinger equation, Comm. Partial Differential Equations, 21 (1996), 1431-1449.

[28]

J. Wang, J. Xu and F. Zhang, Existence of solutions for nonperiodic superquadratic Hamiltonian elliptic systems, Nonlinear Anal., 72 (2010), 1949-1960. doi: 10.1016/j.na.2009.09.035.

[29]

M. Willem, "Minimax Theorems," Birkhäuser, Berlin, 1996.

[30]

J. Yang, Nontrivial solutions of semilinear elliptic systems in $\mathbbR$N, Electron. J. Diff. Eqns., 6 (2001), 343-357.

[31]

F. Zhao, L. Zhao and Y. Ding, Multiple solutions for asymptotically linear elliptic systems, Nonlinear Differ. Equ. Appl., 15 (2008), 673-688.

[32]

F. Zhao, L. Zhao and Y. Ding, Infinitely many solutions for asymptotically linear periodic Hamiltonian ellitpic systems, ESAIM: Control, Optimisation and Calculus of Variations, 16 (2010), 77-91. doi: 10.1051/cocv:2008064.

show all references

References:
[1]

N. Ackermann, On a periodic Schrödinger equation with nonlinear superlinear part, Math. Z., 248 (2004), 423-443. doi: 10.1007/s00209-004-0663-y.

[2]

N. Ackermann, A superposition principle and multibump solutions of periodic Schrödinger equations, J. Func. Anal., 234 (2006), 277-320. doi: 10.1016/j.jfa.2005.11.010.

[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), 673-690. doi: 10.1016/S0022-247X(02)00413-4.

[4]

A. I. Ávila and J. Yang, Multiple solutions of nonlinear elliptic systems, Nonlinear Differ. Equ. Appl., 12 (2005), 459-479.

[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), 348-376.

[6]

T. Bartsch and D. G. De Figueiredo, Infinitely many solutions of nonlinear elliptic systems, in "Progress in Nonlinear Differential Equations and Their Applications," 35, Birkhäuser, Basel/Switzerland, (1999), 51-67.

[7]

T. Bartsch and Y. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Math. Nach., 279 (2006), 1-22. doi: 10.1002/mana.200410420.

[8]

V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals, Inven. Math., 52 (1979), 241-273. doi: 10.1007/BF01389883.

[9]

V. Coti-Zelati and P. H. Rabinowitz, Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. Amer. Math. Soc., 4 (1991), 693-727. doi: 10.1090/S0894-0347-1991-1119200-3.

[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), 1217-1269. doi: 10.1002/cpa.3160451002.

[11]

D. G. De Figueiredo and Y. H. Ding, Strongly indefinite functionals and multiple solutions of elliptic systems, Tran. Amer. Math. Soc., 355 (2003), 2973-2989. doi: 10.1090/S0002-9947-03-03257-4.

[12]

D. G. De Figueiredo and P. L. Felmer, On superquadratic elliptic systems, Tran. Amer. Math. Soc., 343 (1994), 97-116.

[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), 471-496. doi: 10.1016/j.jfa.2004.09.008.

[14]

D. G. De Figueiredo and J. Yang, Decay, symmetry and existence of solutions of semilinear elliptic systems, Nonlinear Anal., 33 (1998), 211-234. doi: 10.1016/S0362-546X(97)00548-8.

[15]

Y. Ding, "Variational Methods for Strongly Indefinite Problems," Interdisciplinary Mathematical Sciences, 7, World Sci., 2007. doi: 10.1142/9789812709639.

[16]

Y. Ding and L. Jeanjean, Homoclinic orbits for a non periodic Hamiltonian system, J. Differential Equations, 237 (2007), 473-490. doi: 10.1016/j.jde.2007.03.005.

[17]

Y. Ding and C. Lee, Existence and exponential decay of homoclinics in a nonperiodic superquadratic Hamiltonian system, J. Differential Equations, 246 (2009), 2829-2848.

[18]

J. Hulshof and R. C. A. M. Van de Vorst, Differential systems with strongly variational structure, J. Func. Anal., 114 (1993), 32-58. doi: 10.1006/jfan.1993.1062.

[19]

W. Kryszewski and A. Szulkin, An infinite dimensional Morse theory with applications, Tran. Amer. Math. Soc., 349 (1997), 3181-3234. doi: 10.1090/S0002-9947-97-01963-6.

[20]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equations, Adv. Differential Equations, 3 (1998), 441-472.

[21]

G. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Comm. Contemp. Math., 4 (2002), 763-776. doi: 10.1142/S0219199702000853.

[22]

G. Li and J. Yang, Asymptotically linear elliptic systems, Comm. Partial Differential Equations, 29 (2004), 925-954.

[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), 160-176.

[24]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics, IV Analysis of Operators," Academic Press, New York, 1978.

[25]

E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian stysems, Math. Z., 209 (1992), 133-160.

[26]

B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $R$N, Adv. Differential Equations, 5 (2000), 1445-1464.

[27]

C. Troestler and M. Willem, Nontrivial solution of a semilinear Schrödinger equation, Comm. Partial Differential Equations, 21 (1996), 1431-1449.

[28]

J. Wang, J. Xu and F. Zhang, Existence of solutions for nonperiodic superquadratic Hamiltonian elliptic systems, Nonlinear Anal., 72 (2010), 1949-1960. doi: 10.1016/j.na.2009.09.035.

[29]

M. Willem, "Minimax Theorems," Birkhäuser, Berlin, 1996.

[30]

J. Yang, Nontrivial solutions of semilinear elliptic systems in $\mathbbR$N, Electron. J. Diff. Eqns., 6 (2001), 343-357.

[31]

F. Zhao, L. Zhao and Y. Ding, Multiple solutions for asymptotically linear elliptic systems, Nonlinear Differ. Equ. Appl., 15 (2008), 673-688.

[32]

F. Zhao, L. Zhao and Y. Ding, Infinitely many solutions for asymptotically linear periodic Hamiltonian ellitpic systems, ESAIM: Control, Optimisation and Calculus of Variations, 16 (2010), 77-91. doi: 10.1051/cocv:2008064.

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