# American Institute of Mathematical Sciences

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.

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##### 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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