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Existence and concentration of semiclassical solutions for Hamiltonian elliptic system
1. | School of Mathematics and Statistics, Hunan University of Commerce, Changsha, 410205 Hunan, China, China |
2. | School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China |
References:
[1] |
N. Ackermann, A nonlinear superposition principle and multibump solution of periodic Schrödinger equations,, \emph{J. Funct. Anal.}, 234 (2006), 423.
doi: 10.1016/j.jfa.2005.11.010. |
[2] |
A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations,, \emph{Arch. Ration. Mech. Anal.}, 140 (1997), 285.
doi: 10.1007/s002050050067. |
[3] |
A. I. Ávila and J. Yang, On the existence and shape of least energy solutions for some elliptic systems,, \emph{J. Differential Equations}, 191 (2003), 348.
doi: 10.1016/S0022-0396(03)00017-2. |
[4] |
A. I. Ávila and J. Yang, Multiple solutions of nonlinear elliptic systems,, \emph{Nonlinear Differ. Equ. Appl.}, 12 (2005), 459.
doi: 10.1007/s00030-005-0022-7. |
[5] |
T. Bartsch and D. G. De Figueiredo, Infinitely mang solutions of nonlinear elliptic systems,, in: Progr. Nonlinear Differential Equations Appl. Vol. 35, (1999), 51.
|
[6] |
T. Bartsch and Y. H. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory,, \emph{Math. Nach.}, 279 (2006), 1267.
doi: 10.1002/mana.200410420. |
[7] |
J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity,, \emph{Arch. Ration. Mech. Anal.}, 185 (2007), 185. Google Scholar |
[8] |
J. Byeon and Z. Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, II,, \emph{Calc. Var. Part. Diffe. Equ.}, 18 (2003), 207.
doi: 10.1007/s00526-002-0191-8. |
[9] |
D. G. De Figueiredo and J. Yang, Decay, symmetry and existence of solutions of semilinear elliptic systems,, \emph{Nonlinear. Anal.}, 331 (1998), 211.
doi: 10.1016/S0362-546X(97)00548-8. |
[10] |
Y. H. Ding, Variational Methods for Strongly Indefinite Problems,, World Scientific Press, (2008).
doi: 10.1142/9789812709639. |
[11] |
Y. H. Ding, Semi-classical ground states concentrating on the nonlinear potential for a Dirac equation,, \emph{J. Differential Equations}, 249 (2010), 1015.
doi: 10.1016/j.jde.2010.03.022. |
[12] |
Y. H. Ding and X. Y, Liu, Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities,, \emph{Manuscr. Math.}, 140 (2013), 51.
doi: 10.1007/s00229-011-0530-1. |
[13] |
Y. H. Ding and X. Y. Liu, Semi-classical limits of ground states of a nonlinear Dirac equation,, \emph{J. Differential Equations}, 252 (2012), 4962.
doi: 10.1016/j.jde.2012.01.023. |
[14] |
Y. H. Ding and X. Y. Liu, On Semiclassical ground states of a nonlinear Dirac equation,, \emph{Rev. Math. Phys.}, 24 (2012).
doi: 10.1142/S0129055X12500298. |
[15] |
Y. H. Ding, C. Lee and B. Ruf, On semiclassical states of a nonlinear Dirac equation,, \emph{Proc. Roy. Soc. Edinb. A}, 143 (2013), 765.
doi: 10.1017/S0308210511001752. |
[16] |
Y. H. Ding, C. Lee and F. K. Zhao, Semiclassical limits of ground state solutions to Schrödinger systems,, \emph{Calc. Var. Partial Differential Equations}, 51 (2014), 725.
doi: 10.1007/s00526-013-0693-6. |
[17] |
Y. H. Ding and J. C. Wei, Stationary states of nonlinear Dirac equations with general potentials,, \emph{Rev. Math. Phys.}, 20 (2008), 1007.
doi: 10.1142/S0129055X0800350X. |
[18] |
M. Esteban and E. Séré, Stationary states of the nonlinear Dirac equation: A variational approach,, \emph{Comm. Math. Phys.}, 171 (1995), 250.
|
[19] |
A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential,, \emph{J. Funct. Anal.}, 69 (1986), 397.
doi: 10.1016/0022-1236(86)90096-0. |
[20] |
C. F. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method,, \emph{Commun. Part. Diffe. Equ.}, 21 (1996), 787.
doi: 10.1080/03605309608821208. |
[21] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer, (1983).
doi: 10.1007/978-3-642-61798-0. |
[22] |
S. Y. He, R. M. Zhang and F. K. Zhao, A note on a superlinear and periodic elliptic system in the whole space,, \emph{Comm. Pure. Appl. Anal.}, 10 (2011), 1149.
doi: 10.3934/cpaa.2011.10.1149. |
[23] |
L. Jeanjean and K. Tanaka, Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities,, \emph{Calc. Var. Part. Diffe. Equ.}, 21 (2004), 287.
doi: 10.1007/s00526-003-0261-6. |
[24] |
Y. Y. Li, On singularly perturbed elliptic equation,, \emph{Adv. Diff. Eqns}, 2 (1997), 955.
|
[25] |
G. Li and J. Yang, Asymptotically linear elliptic systems,, \emph{Commun. Part. Diffe. Equ.}, 29 (2004), 925.
doi: 10.1081/PDE-120037337. |
[26] |
P. L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case Part II,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 1 (1984), 223.
|
[27] |
F. Liao, X. H. Tang and J. Zhang, Existence of solutions for periodic elliptic system with general superlinear nonlinearity,, \emph{Z. Angew. Math. Phys.}, 66 (2015), 689.
doi: 10.1007/s00033-014-0425-6. |
[28] |
Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential,, \emph{Commun. Math. Phys.}, 131 (1990), 223.
|
[29] |
M. del Pino and P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method,, \emph{Math. Ann.}, 324 (2002), 1.
doi: 10.1007/s002080200327. |
[30] |
B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $\mathbbR^N$,, \emph{Adv. Differential Equations}, 5 (2000), 1445.
|
[31] |
A. Szulkin and T. Weth, Ground state solutions for some indefinite problems,, \emph{J. Funct. Anal.}, 257 (2009), 3802.
doi: 10.1016/j.jfa.2009.09.013. |
[32] |
X. H. Tang, Ground state solutions of Nehari-Pankov type for a superlinear Hamiltonian elliptic system on $\mathbbR^N$,, \emph{Canad. Math. Bull.}, 58 (2015), 651.
doi: 10.4153/CMB-2015-019-2. |
[33] |
X. F. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, \emph{Comm. Math. Phys.}, 153 (1993), 229.
|
[34] |
J. Wang, J. X. Xu and F. B. Zhang, Existence of semiclassical ground-state solutions for semilinear elliptic systems,, \emph{Pro. Royal Soci. Edinb: Sec. A}, 142 (2012), 867.
doi: 10.1017/S0308210511000254. |
[35] |
L. R. Xia, J. Zhang and F. K. Zhao, Ground state solutions for superlinear elliptic systems on $\mathbbR^N$,, \emph{J. Math. Anal. Appl.}, 401 (2013), 518.
doi: 10.1016/j.jmaa.2012.12.041. |
[36] |
M. B. Yang, W. X. Chen and Y. H. Ding, Solutions of a class of Hamiltonian elliptic systems in $\mathbbR^N$,, \emph{J. Math. Anal. Appl.}, 352 (2010), 338.
doi: 10.1016/j.jmaa.2009.07.052. |
[37] |
F. K. Zhao, L. G. Zhao and Y. H. Ding, Infinitly mang solutions for asymptotically linear periodic Hamiltonian system,, \emph{ESAIM: Control, 16 (2010), 77.
doi: 10.1051/cocv:2008064. |
[38] |
F. K. Zhao, L. G. Zhao and Y. H. Ding, Multiple solutions for asympototically linear elliptic systems,, \emph{NoDEA Nonlinear Differential Equations Appl.}, 15 (2008), 673.
doi: 10.1007/s00030-008-7080-6. |
[39] |
F. K. Zhao, L. G. Zhao and Y. H. Ding, Multiple solution for a superlinear and periodic ellipic system on $\mathbbR^N$,, \emph{Z. Angew. Math. Phys.}, 62 (2011), 495. Google Scholar |
[40] |
F. K. Zhao, L. G. Zhao and Y. H. Ding, A note on superlinear Hamiltonian elliptic systems,, \emph{J. Math. Phy.}, 50 (2009).
doi: 10.1063/1.3256120. |
[41] |
F. K. Zhao and Y. H. Ding, On Hamiltonian elliptic systems with periodic or non-periodic potentials,, \emph{J. Differential Equations}, 249 (2010), 2964.
doi: 10.1016/j.jde.2010.09.014. |
[42] |
R. M. Zhang, J. Chen and F. K. Zhao, Multiple solutions for superlinear elliptic systems of Hamiltonian type,, \emph{Disc. Contin. Dyn. Syst. Ser. A}, 30 (2011), 1249.
doi: 10.3934/dcds.2011.30.1249. |
[43] |
J. Zhang, W. P. Qin and F. K. Zhao, Existence and multiplicity of solutions for asymptotically linear nonperiodic Hamiltonian elliptic system,, \emph{J. Math. Anal. Appl.}, 399 (2013), 433.
doi: 10.1016/j.jmaa.2012.10.030. |
[44] |
J. Zhang, X. H. Tang and W. Zhang, Ground-state solutions for superquadratic Hamiltonian elliptic systems with gradient terms,, \emph{Nonlinear Anal.}, 95 (2014), 1.
doi: 10.1016/j.na.2013.07.027. |
[45] |
J. Zhang, X. H. Tang and W. Zhang, Semiclassical solutions for a class of Schrödinger system with magnetic potentials,, \emph{J. Math. Anal. Appl.}, 414 (2014), 357.
doi: 10.1016/j.jmaa.2013.12.060. |
[46] |
J. Zhang, X. H. Tang and W. Zhang, On semiclassical ground state solutions for Hamiltonian elliptic systems,, \emph{Appl. Anal.}, 94 (2015), 1380.
doi: 10.1080/00036811.2014.931940. |
[47] |
W. Zhang, X. H. Tang and J. Zhang, Ground state solutions for a diffusion system,, \emph{Comput. Math. Appl.}, 69 (2015), 337.
doi: 10.1016/j.camwa.2014.12.012. |
[48] |
M. Willem, Minimax Theorems,, Birkh\, (1996). Google Scholar |
show all references
References:
[1] |
N. Ackermann, A nonlinear superposition principle and multibump solution of periodic Schrödinger equations,, \emph{J. Funct. Anal.}, 234 (2006), 423.
doi: 10.1016/j.jfa.2005.11.010. |
[2] |
A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations,, \emph{Arch. Ration. Mech. Anal.}, 140 (1997), 285.
doi: 10.1007/s002050050067. |
[3] |
A. I. Ávila and J. Yang, On the existence and shape of least energy solutions for some elliptic systems,, \emph{J. Differential Equations}, 191 (2003), 348.
doi: 10.1016/S0022-0396(03)00017-2. |
[4] |
A. I. Ávila and J. Yang, Multiple solutions of nonlinear elliptic systems,, \emph{Nonlinear Differ. Equ. Appl.}, 12 (2005), 459.
doi: 10.1007/s00030-005-0022-7. |
[5] |
T. Bartsch and D. G. De Figueiredo, Infinitely mang solutions of nonlinear elliptic systems,, in: Progr. Nonlinear Differential Equations Appl. Vol. 35, (1999), 51.
|
[6] |
T. Bartsch and Y. H. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory,, \emph{Math. Nach.}, 279 (2006), 1267.
doi: 10.1002/mana.200410420. |
[7] |
J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity,, \emph{Arch. Ration. Mech. Anal.}, 185 (2007), 185. Google Scholar |
[8] |
J. Byeon and Z. Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, II,, \emph{Calc. Var. Part. Diffe. Equ.}, 18 (2003), 207.
doi: 10.1007/s00526-002-0191-8. |
[9] |
D. G. De Figueiredo and J. Yang, Decay, symmetry and existence of solutions of semilinear elliptic systems,, \emph{Nonlinear. Anal.}, 331 (1998), 211.
doi: 10.1016/S0362-546X(97)00548-8. |
[10] |
Y. H. Ding, Variational Methods for Strongly Indefinite Problems,, World Scientific Press, (2008).
doi: 10.1142/9789812709639. |
[11] |
Y. H. Ding, Semi-classical ground states concentrating on the nonlinear potential for a Dirac equation,, \emph{J. Differential Equations}, 249 (2010), 1015.
doi: 10.1016/j.jde.2010.03.022. |
[12] |
Y. H. Ding and X. Y, Liu, Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities,, \emph{Manuscr. Math.}, 140 (2013), 51.
doi: 10.1007/s00229-011-0530-1. |
[13] |
Y. H. Ding and X. Y. Liu, Semi-classical limits of ground states of a nonlinear Dirac equation,, \emph{J. Differential Equations}, 252 (2012), 4962.
doi: 10.1016/j.jde.2012.01.023. |
[14] |
Y. H. Ding and X. Y. Liu, On Semiclassical ground states of a nonlinear Dirac equation,, \emph{Rev. Math. Phys.}, 24 (2012).
doi: 10.1142/S0129055X12500298. |
[15] |
Y. H. Ding, C. Lee and B. Ruf, On semiclassical states of a nonlinear Dirac equation,, \emph{Proc. Roy. Soc. Edinb. A}, 143 (2013), 765.
doi: 10.1017/S0308210511001752. |
[16] |
Y. H. Ding, C. Lee and F. K. Zhao, Semiclassical limits of ground state solutions to Schrödinger systems,, \emph{Calc. Var. Partial Differential Equations}, 51 (2014), 725.
doi: 10.1007/s00526-013-0693-6. |
[17] |
Y. H. Ding and J. C. Wei, Stationary states of nonlinear Dirac equations with general potentials,, \emph{Rev. Math. Phys.}, 20 (2008), 1007.
doi: 10.1142/S0129055X0800350X. |
[18] |
M. Esteban and E. Séré, Stationary states of the nonlinear Dirac equation: A variational approach,, \emph{Comm. Math. Phys.}, 171 (1995), 250.
|
[19] |
A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential,, \emph{J. Funct. Anal.}, 69 (1986), 397.
doi: 10.1016/0022-1236(86)90096-0. |
[20] |
C. F. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method,, \emph{Commun. Part. Diffe. Equ.}, 21 (1996), 787.
doi: 10.1080/03605309608821208. |
[21] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer, (1983).
doi: 10.1007/978-3-642-61798-0. |
[22] |
S. Y. He, R. M. Zhang and F. K. Zhao, A note on a superlinear and periodic elliptic system in the whole space,, \emph{Comm. Pure. Appl. Anal.}, 10 (2011), 1149.
doi: 10.3934/cpaa.2011.10.1149. |
[23] |
L. Jeanjean and K. Tanaka, Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities,, \emph{Calc. Var. Part. Diffe. Equ.}, 21 (2004), 287.
doi: 10.1007/s00526-003-0261-6. |
[24] |
Y. Y. Li, On singularly perturbed elliptic equation,, \emph{Adv. Diff. Eqns}, 2 (1997), 955.
|
[25] |
G. Li and J. Yang, Asymptotically linear elliptic systems,, \emph{Commun. Part. Diffe. Equ.}, 29 (2004), 925.
doi: 10.1081/PDE-120037337. |
[26] |
P. L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case Part II,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 1 (1984), 223.
|
[27] |
F. Liao, X. H. Tang and J. Zhang, Existence of solutions for periodic elliptic system with general superlinear nonlinearity,, \emph{Z. Angew. Math. Phys.}, 66 (2015), 689.
doi: 10.1007/s00033-014-0425-6. |
[28] |
Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential,, \emph{Commun. Math. Phys.}, 131 (1990), 223.
|
[29] |
M. del Pino and P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method,, \emph{Math. Ann.}, 324 (2002), 1.
doi: 10.1007/s002080200327. |
[30] |
B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $\mathbbR^N$,, \emph{Adv. Differential Equations}, 5 (2000), 1445.
|
[31] |
A. Szulkin and T. Weth, Ground state solutions for some indefinite problems,, \emph{J. Funct. Anal.}, 257 (2009), 3802.
doi: 10.1016/j.jfa.2009.09.013. |
[32] |
X. H. Tang, Ground state solutions of Nehari-Pankov type for a superlinear Hamiltonian elliptic system on $\mathbbR^N$,, \emph{Canad. Math. Bull.}, 58 (2015), 651.
doi: 10.4153/CMB-2015-019-2. |
[33] |
X. F. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, \emph{Comm. Math. Phys.}, 153 (1993), 229.
|
[34] |
J. Wang, J. X. Xu and F. B. Zhang, Existence of semiclassical ground-state solutions for semilinear elliptic systems,, \emph{Pro. Royal Soci. Edinb: Sec. A}, 142 (2012), 867.
doi: 10.1017/S0308210511000254. |
[35] |
L. R. Xia, J. Zhang and F. K. Zhao, Ground state solutions for superlinear elliptic systems on $\mathbbR^N$,, \emph{J. Math. Anal. Appl.}, 401 (2013), 518.
doi: 10.1016/j.jmaa.2012.12.041. |
[36] |
M. B. Yang, W. X. Chen and Y. H. Ding, Solutions of a class of Hamiltonian elliptic systems in $\mathbbR^N$,, \emph{J. Math. Anal. Appl.}, 352 (2010), 338.
doi: 10.1016/j.jmaa.2009.07.052. |
[37] |
F. K. Zhao, L. G. Zhao and Y. H. Ding, Infinitly mang solutions for asymptotically linear periodic Hamiltonian system,, \emph{ESAIM: Control, 16 (2010), 77.
doi: 10.1051/cocv:2008064. |
[38] |
F. K. Zhao, L. G. Zhao and Y. H. Ding, Multiple solutions for asympototically linear elliptic systems,, \emph{NoDEA Nonlinear Differential Equations Appl.}, 15 (2008), 673.
doi: 10.1007/s00030-008-7080-6. |
[39] |
F. K. Zhao, L. G. Zhao and Y. H. Ding, Multiple solution for a superlinear and periodic ellipic system on $\mathbbR^N$,, \emph{Z. Angew. Math. Phys.}, 62 (2011), 495. Google Scholar |
[40] |
F. K. Zhao, L. G. Zhao and Y. H. Ding, A note on superlinear Hamiltonian elliptic systems,, \emph{J. Math. Phy.}, 50 (2009).
doi: 10.1063/1.3256120. |
[41] |
F. K. Zhao and Y. H. Ding, On Hamiltonian elliptic systems with periodic or non-periodic potentials,, \emph{J. Differential Equations}, 249 (2010), 2964.
doi: 10.1016/j.jde.2010.09.014. |
[42] |
R. M. Zhang, J. Chen and F. K. Zhao, Multiple solutions for superlinear elliptic systems of Hamiltonian type,, \emph{Disc. Contin. Dyn. Syst. Ser. A}, 30 (2011), 1249.
doi: 10.3934/dcds.2011.30.1249. |
[43] |
J. Zhang, W. P. Qin and F. K. Zhao, Existence and multiplicity of solutions for asymptotically linear nonperiodic Hamiltonian elliptic system,, \emph{J. Math. Anal. Appl.}, 399 (2013), 433.
doi: 10.1016/j.jmaa.2012.10.030. |
[44] |
J. Zhang, X. H. Tang and W. Zhang, Ground-state solutions for superquadratic Hamiltonian elliptic systems with gradient terms,, \emph{Nonlinear Anal.}, 95 (2014), 1.
doi: 10.1016/j.na.2013.07.027. |
[45] |
J. Zhang, X. H. Tang and W. Zhang, Semiclassical solutions for a class of Schrödinger system with magnetic potentials,, \emph{J. Math. Anal. Appl.}, 414 (2014), 357.
doi: 10.1016/j.jmaa.2013.12.060. |
[46] |
J. Zhang, X. H. Tang and W. Zhang, On semiclassical ground state solutions for Hamiltonian elliptic systems,, \emph{Appl. Anal.}, 94 (2015), 1380.
doi: 10.1080/00036811.2014.931940. |
[47] |
W. Zhang, X. H. Tang and J. Zhang, Ground state solutions for a diffusion system,, \emph{Comput. Math. Appl.}, 69 (2015), 337.
doi: 10.1016/j.camwa.2014.12.012. |
[48] |
M. Willem, Minimax Theorems,, Birkh\, (1996). Google Scholar |
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