Existence and concentration of semiclassical solutions for Hamiltonian elliptic system

Jian  Zhang; Wen  Zhang; Xiaoliang  Xie

doi:10.3934/cpaa.2016.15.599

Previous Article
Non-sharp travelling waves for a dual porous medium equation
CPAA Home
This Issue
Next Article
Center problem for systems with two monomial nonlinearities

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

Existence and concentration of semiclassical solutions for Hamiltonian elliptic system

Jian Zhang ^1, , Wen Zhang ^2, and Xiaoliang Xie ^1,

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

Abstract
Related Papers

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 $\epsilon>0$.

Keywords: concentration, strongly indefinite functionals., semiclassical ground states, Hamiltonian elliptic systems.

Mathematics Subject Classification: 35J50, 58E0.

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:

[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. 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, Birkhäuser, Basel, Switzerland, (1999), 51-67. Google Scholar
[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. Google Scholar
[7]	J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal., 185 (2007), 185-200. Google Scholar
[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. Google Scholar
[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. 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, J. Differential Equations, 249 (2010), 1015-1034. 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, Manuscr. Math., 140 (2013), 51-82. 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, J. Differential Equations, 252 (2012), 4962-4987. 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, Rev. Math. Phys., 24 (2012), 1250029. doi: 10.1142/S0129055X12500298. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	M. Esteban and E. Séré, Stationary states of the nonlinear Dirac equation: A variational approach, Comm. Math. Phys., 171 (1995), 250-323. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. 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, Comm. Pure. Appl. Anal., 10 (2011), 1149-1163. 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, Calc. Var. Part. Diffe. Equ., 21 (2004), 287-318. doi: 10.1007/s00526-003-0261-6. Google Scholar
[24]	Y. Y. Li, On singularly perturbed elliptic equation, Adv. Diff. Eqns, 2 (1997), 955-980. Google Scholar
[25]	G. Li and J. Yang, Asymptotically linear elliptic systems, Commun. Part. Diffe. Equ., 29 (2004), 925-954. 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, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[30]	B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $\mathbbR^N$, Adv. Differential Equations, 5 (2000), 1445-1464. Google Scholar
[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. Google Scholar
[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. Google Scholar
[33]	X. F. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[39]	F. K. Zhao, L. G. Zhao and Y. H. Ding, Multiple solution for a superlinear and periodic ellipic system on $\mathbbR^N$, Z. Angew. Math. Phys., 62 (2011), 495-511. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[48]	M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996. Google Scholar

show all references

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. 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, Birkhäuser, Basel, Switzerland, (1999), 51-67. Google Scholar
[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. Google Scholar
[7]	J. Byeon and L. Jeanjean, Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. Ration. Mech. Anal., 185 (2007), 185-200. Google Scholar
[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. Google Scholar
[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. 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, J. Differential Equations, 249 (2010), 1015-1034. 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, Manuscr. Math., 140 (2013), 51-82. 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, J. Differential Equations, 252 (2012), 4962-4987. 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, Rev. Math. Phys., 24 (2012), 1250029. doi: 10.1142/S0129055X12500298. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[18]	M. Esteban and E. Séré, Stationary states of the nonlinear Dirac equation: A variational approach, Comm. Math. Phys., 171 (1995), 250-323. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. 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, Comm. Pure. Appl. Anal., 10 (2011), 1149-1163. 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, Calc. Var. Part. Diffe. Equ., 21 (2004), 287-318. doi: 10.1007/s00526-003-0261-6. Google Scholar
[24]	Y. Y. Li, On singularly perturbed elliptic equation, Adv. Diff. Eqns, 2 (1997), 955-980. Google Scholar
[25]	G. Li and J. Yang, Asymptotically linear elliptic systems, Commun. Part. Diffe. Equ., 29 (2004), 925-954. 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, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[30]	B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $\mathbbR^N$, Adv. Differential Equations, 5 (2000), 1445-1464. Google Scholar
[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. Google Scholar
[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. Google Scholar
[33]	X. F. Wang, On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153 (1993), 229-244. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[39]	F. K. Zhao, L. G. Zhao and Y. H. Ding, Multiple solution for a superlinear and periodic ellipic system on $\mathbbR^N$, Z. Angew. Math. Phys., 62 (2011), 495-511. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[48]	M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996. Google Scholar

[1]	Xu Zhang. On the concentration of semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5389-5413. doi: 10.3934/dcds.2018238
[2]	Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297
[3]	Claudianor O. Alves, Geilson F. Germano. Existence of ground state solution and concentration of maxima for a class of indefinite variational problems. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2887-2906. doi: 10.3934/cpaa.2020126
[4]	Marcos L. M. Carvalho, Edcarlos D. Silva, Claudiney Goulart, Carlos A. Santos. Ground and bound state solutions for quasilinear elliptic systems including singular nonlinearities and indefinite potentials. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4401-4432. doi: 10.3934/cpaa.2020201
[5]	Yongpeng Chen, Yuxia Guo, Zhongwei Tang. Concentration of ground state solutions for quasilinear Schrödinger systems with critical exponents. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2693-2715. doi: 10.3934/cpaa.2019120
[6]	Jian Zhang, Wen Zhang, Xianhua Tang. Ground state solutions for Hamiltonian elliptic system with inverse square potential. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4565-4583. doi: 10.3934/dcds.2017195
[7]	Jian Zhang, Wen Zhang. Existence and decay property of ground state solutions for Hamiltonian elliptic system. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2433-2455. doi: 10.3934/cpaa.2019110
[8]	Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316
[9]	Qinqin Zhang. Homoclinic orbits for discrete Hamiltonian systems with indefinite linear part. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1929-1940. doi: 10.3934/cpaa.2015.14.1929
[10]	B. Buffoni, F. Giannoni. Brake periodic orbits of prescribed Hamiltonian for indefinite Lagrangian systems. Discrete & Continuous Dynamical Systems, 1995, 1 (2) : 217-222. doi: 10.3934/dcds.1995.1.217
[11]	Hui Zhang, Jun Wang, Fubao Zhang. Semiclassical states for fractional Choquard equations with critical growth. Communications on Pure & Applied Analysis, 2019, 18 (1) : 519-538. doi: 10.3934/cpaa.2019026
[12]	Paolo Caldiroli. Radial and non radial ground states for a class of dilation invariant fourth order semilinear elliptic equations on $R^n$. Communications on Pure & Applied Analysis, 2014, 13 (2) : 811-821. doi: 10.3934/cpaa.2014.13.811
[13]	Rumei Zhang, Jin Chen, Fukun Zhao. Multiple solutions for superlinear elliptic systems of Hamiltonian type. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1249-1262. doi: 10.3934/dcds.2011.30.1249
[14]	Eugenio Montefusco, Benedetta Pellacci, Marco Squassina. Energy convexity estimates for non-degenerate ground states of nonlinear 1D Schrödinger systems. Communications on Pure & Applied Analysis, 2010, 9 (4) : 867-884. doi: 10.3934/cpaa.2010.9.867
[15]	Dongdong Qin, Xianhua Tang, Qingfang Wu. Ground states of nonlinear Schrödinger systems with periodic or non-periodic potentials. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1261-1280. doi: 10.3934/cpaa.2019061
[16]	Wenxiong Chen, Congming Li. Indefinite elliptic problems in a domain. Discrete & Continuous Dynamical Systems, 1997, 3 (3) : 333-340. doi: 10.3934/dcds.1997.3.333
[17]	Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176
[18]	João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138
[19]	Manassés de Souza. On a singular Hamiltonian elliptic systems involving critical growth in dimension two. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1859-1874. doi: 10.3934/cpaa.2012.11.1859
[20]	Daniela Cárcamo-Díaz, Jesús F. Palacián, Claudio Vidal, Patricia Yanguas. Nonlinear stability of elliptic equilibria in hamiltonian systems with exponential time estimates. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5183-5208. doi: 10.3934/dcds.2021073

2020 Impact Factor: 1.916

American Institute of Mathematical Sciences

Existence and concentration of semiclassical solutions for Hamiltonian elliptic system

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Existence and concentration of semiclassical solutions for Hamiltonian elliptic system

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors