# 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 and Applied Analysis, 2016, 15 (2) : 599-622. doi: 10.3934/cpaa.2016.15.599
##### References:
 [1] N. Ackermann, A nonlinear superposition principle and multibump solution of periodic Schrödinger equations, J. Funct. Anal., 234 (2006), 423-443. doi: 10.1016/j.jfa.2005.11.010. [2] A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 140 (1997), 285-300. doi: 10.1007/s002050050067. [3] A. I. Ávila and J. Yang, On the existence and shape of least energy solutions for some elliptic systems, J. Differential Equations, 191 (2003), 348-376. doi: 10.1016/S0022-0396(03)00017-2. [4] A. I. Ávila and J. Yang, Multiple solutions of nonlinear elliptic systems, Nonlinear Differ. Equ. Appl., 12 (2005), 459-479. 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, Birkhäuser, Basel, Switzerland, (1999), 51-67. [6] T. Bartsch and Y. H. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Math. Nach., 279 (2006), 1267-1288. doi: 10.1002/mana.200410420. [7] J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal., 185 (2007), 185-200. [8] J. Byeon and Z. Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, II, Calc. Var. Part. Diffe. Equ., 18 (2003), 207-219. 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, Nonlinear. Anal., 331 (1998), 211-234. 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, J. Differential Equations, 249 (2010), 1015-1034. 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, Manuscr. Math., 140 (2013), 51-82. 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, J. Differential Equations, 252 (2012), 4962-4987. 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, Rev. Math. Phys., 24 (2012), 1250029. doi: 10.1142/S0129055X12500298. [15] Y. H. Ding, C. Lee and B. Ruf, On semiclassical states of a nonlinear Dirac equation, Proc. Roy. Soc. Edinb. A, 143 (2013), 765-790. doi: 10.1017/S0308210511001752. [16] Y. H. Ding, C. Lee and F. K. Zhao, Semiclassical limits of ground state solutions to Schrödinger systems, Calc. Var. Partial Differential Equations, 51 (2014), 725-760. doi: 10.1007/s00526-013-0693-6. [17] Y. H. Ding and J. C. Wei, Stationary states of nonlinear Dirac equations with general potentials, Rev. Math. Phys., 20 (2008), 1007-1032. doi: 10.1142/S0129055X0800350X. [18] M. Esteban and E. Séré, Stationary states of the nonlinear Dirac equation: A variational approach, Comm. Math. Phys., 171 (1995), 250-323. [19] A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0. [20] C. F. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method, Commun. Part. Diffe. Equ., 21 (1996), 787-820. doi: 10.1080/03605309608821208. [21] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 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, Comm. Pure. Appl. Anal., 10 (2011), 1149-1163. doi: 10.3934/cpaa.2011.10.1149. [23] L. Jeanjean and K. Tanaka, Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calc. Var. Part. Diffe. Equ., 21 (2004), 287-318. doi: 10.1007/s00526-003-0261-6. [24] Y. Y. Li, On singularly perturbed elliptic equation, Adv. Diff. Eqns, 2 (1997), 955-980. [25] G. Li and J. Yang, Asymptotically linear elliptic systems, Commun. Part. Diffe. Equ., 29 (2004), 925-954. doi: 10.1081/PDE-120037337. [26] P. L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case Part II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283. [27] F. Liao, X. H. Tang and J. Zhang, Existence of solutions for periodic elliptic system with general superlinear nonlinearity, Z. Angew. Math. Phys., 66 (2015), 689-701. 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, Commun. Math. Phys., 131 (1990), 223-253. [29] M. del Pino and P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1-32. doi: 10.1007/s002080200327. [30] B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $\mathbbR^N$, Adv. Differential Equations, 5 (2000), 1445-1464. [31] 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. [32] X. H. Tang, Ground state solutions of Nehari-Pankov type for a superlinear Hamiltonian elliptic system on $\mathbbR^N$, Canad. Math. Bull., 58 (2015), 651-663. doi: 10.4153/CMB-2015-019-2. [33] X. F. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244. [34] J. Wang, J. X. Xu and F. B. Zhang, Existence of semiclassical ground-state solutions for semilinear elliptic systems, Pro. Royal Soci. Edinb: Sec. A, 142 (2012), 867-895. doi: 10.1017/S0308210511000254. [35] L. R. Xia, J. Zhang and F. K. Zhao, Ground state solutions for superlinear elliptic systems on $\mathbbR^N$, J. Math. Anal. Appl., 401 (2013), 518-525. 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 $\mathbbR^N$, J. Math. Anal. Appl., 352 (2010), 338-349. 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, ESAIM: Control, Optim. Calc. Vari., 16 (2010), 77-91. doi: 10.1051/cocv:2008064. [38] F. K. Zhao, L. G. Zhao and Y. H. Ding, Multiple solutions for asympototically linear elliptic systems, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 673-688. 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 $\mathbbR^N$, Z. Angew. Math. Phys., 62 (2011), 495-511. [40] F. K. Zhao, L. G. Zhao and Y. H. Ding, A note on superlinear Hamiltonian elliptic systems, J. Math. Phy., 50 (2009), 112702. doi: 10.1063/1.3256120. [41] F. K. Zhao and Y. H. Ding, On Hamiltonian elliptic systems with periodic or non-periodic potentials, J. Differential Equations, 249 (2010), 2964-2985. 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, Disc. Contin. Dyn. Syst. Ser. A, 30 (2011), 1249-1262. 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, J. Math. Anal. Appl., 399 (2013), 433-441. 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, Nonlinear Anal., 95 (2014), 1-10. 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, J. Math. Anal. Appl., 414 (2014), 357-371. 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, Appl. Anal., 94 (2015), 1380-1396. doi: 10.1080/00036811.2014.931940. [47] W. Zhang, X. H. Tang and J. Zhang, Ground state solutions for a diffusion system, Comput. Math. Appl., 69 (2015), 337-346. doi: 10.1016/j.camwa.2014.12.012. [48] M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996.

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##### References:
 [1] N. Ackermann, A nonlinear superposition principle and multibump solution of periodic Schrödinger equations, J. Funct. Anal., 234 (2006), 423-443. doi: 10.1016/j.jfa.2005.11.010. [2] A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 140 (1997), 285-300. doi: 10.1007/s002050050067. [3] A. I. Ávila and J. Yang, On the existence and shape of least energy solutions for some elliptic systems, J. Differential Equations, 191 (2003), 348-376. doi: 10.1016/S0022-0396(03)00017-2. [4] A. I. Ávila and J. Yang, Multiple solutions of nonlinear elliptic systems, Nonlinear Differ. Equ. Appl., 12 (2005), 459-479. 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, Birkhäuser, Basel, Switzerland, (1999), 51-67. [6] T. Bartsch and Y. H. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Math. Nach., 279 (2006), 1267-1288. doi: 10.1002/mana.200410420. [7] J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal., 185 (2007), 185-200. [8] J. Byeon and Z. Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, II, Calc. Var. Part. Diffe. Equ., 18 (2003), 207-219. 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, Nonlinear. Anal., 331 (1998), 211-234. 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, J. Differential Equations, 249 (2010), 1015-1034. 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, Manuscr. Math., 140 (2013), 51-82. 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, J. Differential Equations, 252 (2012), 4962-4987. 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, Rev. Math. Phys., 24 (2012), 1250029. doi: 10.1142/S0129055X12500298. [15] Y. H. Ding, C. Lee and B. Ruf, On semiclassical states of a nonlinear Dirac equation, Proc. Roy. Soc. Edinb. A, 143 (2013), 765-790. doi: 10.1017/S0308210511001752. [16] Y. H. Ding, C. Lee and F. K. Zhao, Semiclassical limits of ground state solutions to Schrödinger systems, Calc. Var. Partial Differential Equations, 51 (2014), 725-760. doi: 10.1007/s00526-013-0693-6. [17] Y. H. Ding and J. C. Wei, Stationary states of nonlinear Dirac equations with general potentials, Rev. Math. Phys., 20 (2008), 1007-1032. doi: 10.1142/S0129055X0800350X. [18] M. Esteban and E. Séré, Stationary states of the nonlinear Dirac equation: A variational approach, Comm. Math. Phys., 171 (1995), 250-323. [19] A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408. doi: 10.1016/0022-1236(86)90096-0. [20] C. F. Gui, Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method, Commun. Part. Diffe. Equ., 21 (1996), 787-820. doi: 10.1080/03605309608821208. [21] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 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, Comm. Pure. Appl. Anal., 10 (2011), 1149-1163. doi: 10.3934/cpaa.2011.10.1149. [23] L. Jeanjean and K. Tanaka, Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calc. Var. Part. Diffe. Equ., 21 (2004), 287-318. doi: 10.1007/s00526-003-0261-6. [24] Y. Y. Li, On singularly perturbed elliptic equation, Adv. Diff. Eqns, 2 (1997), 955-980. [25] G. Li and J. Yang, Asymptotically linear elliptic systems, Commun. Part. Diffe. Equ., 29 (2004), 925-954. doi: 10.1081/PDE-120037337. [26] P. L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case Part II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283. [27] F. Liao, X. H. Tang and J. Zhang, Existence of solutions for periodic elliptic system with general superlinear nonlinearity, Z. Angew. Math. Phys., 66 (2015), 689-701. 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, Commun. Math. Phys., 131 (1990), 223-253. [29] M. del Pino and P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324 (2002), 1-32. doi: 10.1007/s002080200327. [30] B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $\mathbbR^N$, Adv. Differential Equations, 5 (2000), 1445-1464. [31] 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. [32] X. H. Tang, Ground state solutions of Nehari-Pankov type for a superlinear Hamiltonian elliptic system on $\mathbbR^N$, Canad. Math. Bull., 58 (2015), 651-663. doi: 10.4153/CMB-2015-019-2. [33] X. F. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244. [34] J. Wang, J. X. Xu and F. B. Zhang, Existence of semiclassical ground-state solutions for semilinear elliptic systems, Pro. Royal Soci. Edinb: Sec. A, 142 (2012), 867-895. doi: 10.1017/S0308210511000254. [35] L. R. Xia, J. Zhang and F. K. Zhao, Ground state solutions for superlinear elliptic systems on $\mathbbR^N$, J. Math. Anal. Appl., 401 (2013), 518-525. 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 $\mathbbR^N$, J. Math. Anal. Appl., 352 (2010), 338-349. 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, ESAIM: Control, Optim. Calc. Vari., 16 (2010), 77-91. doi: 10.1051/cocv:2008064. [38] F. K. Zhao, L. G. Zhao and Y. H. Ding, Multiple solutions for asympototically linear elliptic systems, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 673-688. 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 $\mathbbR^N$, Z. Angew. Math. Phys., 62 (2011), 495-511. [40] F. K. Zhao, L. G. Zhao and Y. H. Ding, A note on superlinear Hamiltonian elliptic systems, J. Math. Phy., 50 (2009), 112702. doi: 10.1063/1.3256120. [41] F. K. Zhao and Y. H. Ding, On Hamiltonian elliptic systems with periodic or non-periodic potentials, J. Differential Equations, 249 (2010), 2964-2985. 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, Disc. Contin. Dyn. Syst. Ser. A, 30 (2011), 1249-1262. 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, J. Math. Anal. Appl., 399 (2013), 433-441. 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, Nonlinear Anal., 95 (2014), 1-10. 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, J. Math. Anal. Appl., 414 (2014), 357-371. 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, Appl. Anal., 94 (2015), 1380-1396. doi: 10.1080/00036811.2014.931940. [47] W. Zhang, X. H. Tang and J. Zhang, Ground state solutions for a diffusion system, Comput. Math. Appl., 69 (2015), 337-346. doi: 10.1016/j.camwa.2014.12.012. [48] M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996.
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