# American Institute of Mathematical Sciences

March  2016, 15(2): 599-622. doi: 10.3934/cpaa.2016.15.599

## 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

Received  August 2015 Revised  November 2015 Published  January 2016

In this paper, we study the following Hamiltonian elliptic system with gradient term \begin{eqnarray} &-\epsilon^{2}\Delta \psi +\epsilon b\cdot \nabla \psi +\psi+V(x)\varphi=K(x)f(|\eta|)\varphi \ \ \hbox{in}~\mathbb{R}^{N},\\ &-\epsilon^{2}\Delta \varphi -\epsilon b\cdot \nabla \varphi +\varphi+V(x)\psi=K(x)f(|\eta|)\psi \ \ \hbox{in}~\mathbb{R}^{N}, \end{eqnarray} where $\eta=(\psi,\varphi):\mathbb{R}^{N}\rightarrow\mathbb{R}^{2}$, $V, K\in C(\mathbb{R}^{N}, \mathbb{R})$, $\epsilon$ is a small positive parameter and $b$ is a constant vector. Suppose that $V(x)$ is sign-changing and has at least one global minimum, and $K(x)$ has at least one global maximum, we prove the existence, exponential decay and concentration phenomena of semiclassical ground state solutions for all sufficiently small $\epsilon>0$.
Citation: Jian Zhang, Wen Zhang, Xiaoliang Xie. Existence and concentration of semiclassical solutions for Hamiltonian elliptic system. Communications on Pure & Applied Analysis, 2016, 15 (2) : 599-622. doi: 10.3934/cpaa.2016.15.599
##### References:
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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. Google Scholar [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. Google Scholar [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. Google Scholar [10] Y. H. Ding, Variational Methods for Strongly Indefinite Problems,, World Scientific Press, (2008). doi: 10.1142/9789812709639. Google Scholar [11] Y. H. 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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. Google Scholar [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. Google Scholar [18] M. Esteban and E. Séré, Stationary states of the nonlinear Dirac equation: A variational approach,, \emph{Comm. Math. Phys.}, 171 (1995), 250. Google Scholar [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. Google Scholar [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. Google Scholar [21] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer, (1983). doi: 10.1007/978-3-642-61798-0. Google Scholar [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. Google Scholar [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. Google Scholar [24] Y. Y. Li, On singularly perturbed elliptic equation,, \emph{Adv. Diff. Eqns}, 2 (1997), 955. Google Scholar [25] G. Li and J. Yang, Asymptotically linear elliptic systems,, \emph{Commun. Part. Diffe. Equ.}, 29 (2004), 925. doi: 10.1081/PDE-120037337. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [30] B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $\mathbbR^N$,, \emph{Adv. Differential Equations}, 5 (2000), 1445. Google Scholar [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. Google Scholar [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. Google Scholar [33] X. F. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, \emph{Comm. Math. Phys.}, 153 (1993), 229. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [48] M. Willem, Minimax Theorems,, Birkh\, (1996). Google Scholar

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##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [10] Y. H. Ding, Variational Methods for Strongly Indefinite Problems,, World Scientific Press, (2008). doi: 10.1142/9789812709639. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] M. Esteban and E. Séré, Stationary states of the nonlinear Dirac equation: A variational approach,, \emph{Comm. Math. Phys.}, 171 (1995), 250. Google Scholar [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. Google Scholar [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. Google Scholar [21] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Springer, (1983). doi: 10.1007/978-3-642-61798-0. Google Scholar [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. Google Scholar [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. Google Scholar [24] Y. Y. Li, On singularly perturbed elliptic equation,, \emph{Adv. Diff. Eqns}, 2 (1997), 955. Google Scholar [25] G. Li and J. Yang, Asymptotically linear elliptic systems,, \emph{Commun. Part. Diffe. Equ.}, 29 (2004), 925. doi: 10.1081/PDE-120037337. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [30] B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $\mathbbR^N$,, \emph{Adv. Differential Equations}, 5 (2000), 1445. Google Scholar [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. Google Scholar [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. Google Scholar [33] X. F. Wang, On concentration of positive bound states of nonlinear Schrödinger equations,, \emph{Comm. Math. Phys.}, 153 (1993), 229. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [48] M. Willem, Minimax Theorems,, Birkh\, (1996). Google Scholar
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