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

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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

Abstract
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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 and 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 $\mathbb{R}^N2$, J. Math. Anal. Appl., 276 (2002), 673-690. doi: 10.1016/S0022-247X(02)00413-4.
[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.
[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.
[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.
[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.
[6]	V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals, Inven. Math., 52 (1979), 241-273. doi: 10.1007/BF01389883.
[7]	V. Coti-Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $\mathbb{R}^N2$, Comm. Pure Appl. Math., 45 (1992), 1217-1269. doi: 10.1002/cpa.3160451002.
[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.
[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.
[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.
[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.
[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.
[13]	L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesmann-Lazer type problem set on $\mathbb{R}^N2$, Proc. Roy. Soc Edinburgh, 129A (1999), 787-809.
[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.
[15]	G. Li and J. Yang, Asymptotically linear elliptic systems, Comm. Partial Diff. Eqns., 29 (2004), 925-954.
[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.
[17]	J. L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications," I, Springer-Berlag, Berlin, 1972.
[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.
[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.
[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.
[21]	M. Reed and B. Simon, "Methods of Modern Mathematical Physics, IV Analysis of Operators," Academic Press, New York, 1978.
[22]	E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian stysems, Math. Z., 209 (1992), 133-160. doi: 10.1007/BF02570817.
[23]	B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $R^N$, Adv. Diff. Eqns., 5 (2000), 1445-1464.
[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.
[25]	H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland, Amsterdam, 1978.
[26]	J. Yang, Nontrivial solutions of semilinear elliptic systems in $\mathbb{R}^N2$, Electron. J. Diff. Eqns., conf. 06 (2001), 343-357.
[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.
[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.
[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.

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 $\mathbb{R}^N2$, J. Math. Anal. Appl., 276 (2002), 673-690. doi: 10.1016/S0022-247X(02)00413-4.
[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.
[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.
[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.
[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.
[6]	V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals, Inven. Math., 52 (1979), 241-273. doi: 10.1007/BF01389883.
[7]	V. Coti-Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on $\mathbb{R}^N2$, Comm. Pure Appl. Math., 45 (1992), 1217-1269. doi: 10.1002/cpa.3160451002.
[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.
[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.
[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.
[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.
[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.
[13]	L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesmann-Lazer type problem set on $\mathbb{R}^N2$, Proc. Roy. Soc Edinburgh, 129A (1999), 787-809.
[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.
[15]	G. Li and J. Yang, Asymptotically linear elliptic systems, Comm. Partial Diff. Eqns., 29 (2004), 925-954.
[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.
[17]	J. L. Lions and E. Magenes, "Non-homogeneous Boundary Value Problems and Applications," I, Springer-Berlag, Berlin, 1972.
[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.
[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.
[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.
[21]	M. Reed and B. Simon, "Methods of Modern Mathematical Physics, IV Analysis of Operators," Academic Press, New York, 1978.
[22]	E. Séré, Existence of infinitely many homoclinic orbits in Hamiltonian stysems, Math. Z., 209 (1992), 133-160. doi: 10.1007/BF02570817.
[23]	B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $R^N$, Adv. Diff. Eqns., 5 (2000), 1445-1464.
[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.
[25]	H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland, Amsterdam, 1978.
[26]	J. Yang, Nontrivial solutions of semilinear elliptic systems in $\mathbb{R}^N2$, Electron. J. Diff. Eqns., conf. 06 (2001), 343-357.
[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.
[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.
[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.

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