American Institute of Mathematical Sciences

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July  2011, 10(4): 1149-1163. doi: 10.3934/cpaa.2011.10.1149

A note on a superlinear and periodic elliptic system in the whole space

Shuying He 1, , Rumei Zhang 2, and  Fukun Zhao 3,

1. 

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

Office of Adult Education, Simao Teacher's College, Simao 665000 Yunnan, China
3. 

Department of Mathematics, Yunnan Normal University, Kunming 650092 Yunnan

Received  April 2010 Revised  December 2010 Published  April 2011

This paper is concerned with the following periodic Hamiltonian elliptic system

$ -\Delta u+V(x)u=g(x,v)$ in $R^N,$

$ -\Delta v+V(x)v=f(x,u)$ in $R^N,$

$ u(x)\to 0$ and $v(x)\to 0$ as $|x|\to\infty,$

where the potential $V$ is periodic and has a positive bound from below, $f(x,t)$ and $g(x,t)$ are periodic in $x$ and superlinear but subcritical in $t$ at infinity. By using generalized Nehari manifold method, existence of a positive ground state solution as well as multiple solutions for odd $f$ and $g$ are obtained.

Keywords: variational method, strongly indefinite functionals., Hamiltonian elliptic system.
Mathematics Subject Classification: Primary: 35J50; Secondary: 35J5.
Citation: Shuying He, Rumei Zhang, Fukun Zhao. A note on a superlinear and periodic elliptic system in the whole space. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1149-1163. doi: 10.3934/cpaa.2011.10.1149
References:
[1]

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.  Google Scholar

[2]

A. I. Ávila and J. Yang, Multiple solutions of nonlinear elliptic systems, Nonlinear Diff. Eqns. Appl., 12 (2005), 459-479. doi: 10.1007/s00030-005-0022-7.  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. Diff. Eqns., 191 (2003), 348-376. doi: 10.1016/S0022-0396(03)00017-2.  Google Scholar

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[5]

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.  Google Scholar

[6]

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

[7]

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.  Google Scholar

[8]

D. G. De Figueiredo and Y. 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.  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), 211-234. doi: 10.1016/S0362-546X(97)00548-8.  Google Scholar

[12]

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

[13]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesmann-Lazer type problem set on $\mathbbR^N$, Proc. Roy. Soc Edinburgh, 129A (1999), 787-809.  Google Scholar

[14]

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.  Google Scholar

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G. Li and J. Yang, Asymptotically linear elliptic systems, Comm. Partial Diff. Eqns., 29 (2004), 925-954.  Google Scholar

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Y. Li, Z. Wang and J. Zeng, Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 829-837. doi: 10.1016/j.anihpc.2006.01.003.  Google Scholar

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P. L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case. Part II, Ann. Inst. H. Poincaré, Analyse non linéaire, 1 (1984), 223-283.  Google Scholar

[19]

A. Pankov, Periodic nonlinear Schröinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287. doi: 10.1007/s00032-005-0047-8.  Google Scholar

[20]

A. Pistoia and M. Ramos, Locating the peaks of the least energy solutions to an elliptic system with Neumann boundary conditions, J. Diff. Eqns., 201 (2004), 160-176. doi: 10.1016/j.jde.2004.02.003.  Google Scholar

[21]

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

[22]

E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian stysems, Math. Z., 209 (1992), 133-160. doi: 10.1007/BF02570817.  Google Scholar

[23]

B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $R^N$, Adv. Diff. Eqns., 5 (2000), 1445-1464.  Google Scholar

[24]

A. Szulkin and T. Weth, Ground state solutions for some indefinite problems, J. Funct. Anal., 257 (2009), 3802-3822. doi: 10.1016/j.jfa.2009.09.013.  Google Scholar

[25]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland, Amsterdam, 1978.  Google Scholar

[26]

J. Yang, Nontrivial solutions of semilinear elliptic systems in $\mathbbR^N$, Electron. J. Diff. Eqns., conf. 06 (2001), 343-357.  Google Scholar

[27]

F. Zhao, L. Zhao and Y. Ding, Multiple solutions for asymptotically linear elliptic systems, Nonlinear Differ. Equ. Appl., 15 (2008), 673-688. doi: 10.1007/s00030-008-7080-6.  Google Scholar

[28]

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

[29]

F. Zhao, L. Zhao and Y. Ding, A note on superlinear Hamiltonian elliptic systems, J. Math. Phy., 50 (2009), 112702. doi: 10.1063/1.3256120.  Google Scholar

show all references

References:
[1]

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.  Google Scholar

[2]

A. I. Ávila and J. Yang, Multiple solutions of nonlinear elliptic systems, Nonlinear Diff. Eqns. Appl., 12 (2005), 459-479. doi: 10.1007/s00030-005-0022-7.  Google Scholar

[3]

A. I. Ávila and J. Yang, On the existence and shape of least energy solutions for some elliptic systems, J. Diff. Eqns., 191 (2003), 348-376. doi: 10.1016/S0022-0396(03)00017-2.  Google Scholar

[4]

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

[5]

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.  Google Scholar

[6]

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

[7]

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.  Google Scholar

[8]

D. G. De Figueiredo and Y. 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.  Google Scholar

[9]

D. G. De Figueiredo and P. L. Felmer, On superquadratic elliptic systems, Tran. Amer. Math. Soc., 343 (1994), 97-116. doi: 10.1090/S0002-9947-1994-1214781-2.  Google Scholar

[10]

D. G. De Figueiredo, J. M. 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.  Google Scholar

[11]

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.  Google Scholar

[12]

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

[13]

L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesmann-Lazer type problem set on $\mathbbR^N$, Proc. Roy. Soc Edinburgh, 129A (1999), 787-809.  Google Scholar

[14]

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.  Google Scholar

[15]

G. Li and J. Yang, Asymptotically linear elliptic systems, Comm. Partial Diff. Eqns., 29 (2004), 925-954.  Google Scholar

[16]

Y. Li, Z. Wang and J. Zeng, Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 829-837. doi: 10.1016/j.anihpc.2006.01.003.  Google Scholar

[17]

J. L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications," I, Springer-Berlag, Berlin, 1972.  Google Scholar

[18]

P. L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case. Part II, Ann. Inst. H. Poincaré, Analyse non linéaire, 1 (1984), 223-283.  Google Scholar

[19]

A. Pankov, Periodic nonlinear Schröinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287. doi: 10.1007/s00032-005-0047-8.  Google Scholar

[20]

A. Pistoia and M. Ramos, Locating the peaks of the least energy solutions to an elliptic system with Neumann boundary conditions, J. Diff. Eqns., 201 (2004), 160-176. doi: 10.1016/j.jde.2004.02.003.  Google Scholar

[21]

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

[22]

E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian stysems, Math. Z., 209 (1992), 133-160. doi: 10.1007/BF02570817.  Google Scholar

[23]

B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $R^N$, Adv. Diff. Eqns., 5 (2000), 1445-1464.  Google Scholar

[24]

A. Szulkin and T. Weth, Ground state solutions for some indefinite problems, J. Funct. Anal., 257 (2009), 3802-3822. doi: 10.1016/j.jfa.2009.09.013.  Google Scholar

[25]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland, Amsterdam, 1978.  Google Scholar

[26]

J. Yang, Nontrivial solutions of semilinear elliptic systems in $\mathbbR^N$, Electron. J. Diff. Eqns., conf. 06 (2001), 343-357.  Google Scholar

[27]

F. Zhao, L. Zhao and Y. Ding, Multiple solutions for asymptotically linear elliptic systems, Nonlinear Differ. Equ. Appl., 15 (2008), 673-688. doi: 10.1007/s00030-008-7080-6.  Google Scholar

[28]

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

[29]

F. Zhao, L. Zhao and Y. Ding, A note on superlinear Hamiltonian elliptic systems, J. Math. Phy., 50 (2009), 112702. doi: 10.1063/1.3256120.  Google Scholar

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