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 & 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

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

[1]

Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036

[2]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[3]

Andrea Braides, Antonio Tribuzio. Perturbed minimizing movements of families of functionals. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 373-393. doi: 10.3934/dcdss.2020324

[4]

Kimie Nakashima. Indefinite nonlinear diffusion problem in population genetics. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3837-3855. doi: 10.3934/dcds.2020169

[5]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[6]

Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127

[7]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297

[8]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

[9]

Andrew D. Lewis. Erratum for "nonholonomic and constrained variational mechanics". Journal of Geometric Mechanics, 2020, 12 (4) : 671-675. doi: 10.3934/jgm.2020033

[10]

Simon Hochgerner. Symmetry actuated closed-loop Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 641-669. doi: 10.3934/jgm.2020030

[11]

Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185

[12]

Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020406

[13]

Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2021001

[14]

Philippe G. Ciarlet, Liliana Gratie, Cristinel Mardare. Intrinsic methods in elasticity: a mathematical survey. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 133-164. doi: 10.3934/dcds.2009.23.133

[15]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[16]

Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $ \mathbb{R}^{N} $. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021006

[17]

Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020170

[18]

Chun Liu, Huan Sun. On energetic variational approaches in modeling the nematic liquid crystal flows. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 455-475. doi: 10.3934/dcds.2009.23.455

[19]

Adrian Viorel, Cristian D. Alecsa, Titus O. Pinţa. Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020407

[20]

Philippe Laurençot, Christoph Walker. Variational solutions to an evolution model for MEMS with heterogeneous dielectric properties. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 677-694. doi: 10.3934/dcdss.2020360

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (53)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]