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September  2019, 18(5): 2433-2455. doi: 10.3934/cpaa.2019110

Existence and decay property of ground state solutions for Hamiltonian elliptic system

1. 

School of Mathematics and Statistics, Hunan University of Commerce, Changsha, 410205 Hunan, China

2. 

School of Mathematics and Statistics, Central South University, Changsha, 410083 Hunan, China

* Corresponding author

Received  May 2018 Revised  December 2018 Published  April 2019

Fund Project: This work was supported by the NNSF (Nos. 11701173, 11601145, 11571370, 61772196), by the Natural Science Foundation of Hunan Province (Nos. 2017JJ3130, 2017JJ3131), by the Excellent youth project of Education Department of Hunan Province (17B143), by the Hunan University of Commerce Innovation Driven Project for Young Teacher (16QD008), and by the Project funded by China Postdoctoral Science Foundation (2018M640758).

In this paper we study the following nonlinear Hamiltonian elliptic system with gradient term
$ \begin{eqnarray*} \left\{ \begin{array}{ll} -\Delta u +\vec{b}(x)\cdot \nabla u +u+V(x)v = f(x, |z|)v, \; \; x\in\mathbb{R}^{N}, \\ -\Delta v -\vec{b}(x)\cdot \nabla v +v+V(x)u = f(x, |z|)u, \; \; x\in\mathbb{R}^{N}, \ \end{array} \right. \end{eqnarray*} $
where
$ z = (u, v)\in\mathbb{R}^{2} $
. Under some suitable conditions on the potential and nonlinearity, we obtain the existence of ground state solutions in periodic case and asymptotically periodic case via variational methods, respectively. Moreover, we also explore some properties of these ground state solutions, such as compactness of set of ground state solutions and exponential decay of ground state solutions.
Citation: Jian Zhang, Wen Zhang. Existence and decay property of ground state solutions for Hamiltonian elliptic system. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2433-2455. doi: 10.3934/cpaa.2019110
References:
[1]

C. O. AlvesJ. M. do Ó and O. H. Miyagaki, On nonlinear perturbation of a periodic elliptic problem in $\mathbb{R}^2$ involving critical growth, Nonlinear Anal., 56 (2004), 781-791.  doi: 10.1016/j.na.2003.06.003.

[2]

S. Alama and Y. Y. Li, On "multibump" bound states for certain semilinear elliptic equations, Indiana Univ. Math. J., 41 (1992), 983-1026.  doi: 10.1512/iumj.1992.41.41052.

[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]

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.

[5]

V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math., 52 (1979), 241-273.  doi: 10.1007/BF01389883.

[6] Y. H. Ding, Variational Methods for Strongly Indefinite Problems, World Scientific Press, 2008.  doi: 10.1142/9789812709639.
[7]

D. G. De Figueiredo and Y. H. Ding, Strongly indefinite functions and multiple solutions of elliptic systems, Trans. Am. Math. Soc., 355 (2003), 2973-2989.  doi: 10.1090/S0002-9947-03-03257-4.

[8]

D. G. De Figueiredo and P. L. Felmer, On superquadiatic elliptic systems, Trans. Am. Math. Soc., 343 (1994), 97-116.  doi: 10.2307/2154523.

[9]

D. G. De Figueiredo, Semilinear elliptic systems: existence, multiplicity, symmetry of solutions, Handbook of Differential Equations Stationary Partial Differential Equations, 5, Elsevier, 2008, p.1–48. Chapter1. doi: 10.1016/S1874-5733(08)80008-3.

[10]

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.

[11]

M. J. Esteban and E. Séré, Stationary states of nonlinear Dirac equations: A variational approach, Comm. Math. Phys., 171 (1995), 323-350. 

[12]

J. Hulshof and R. C. A. M. De Vorst, Differential systems with strongly variational structure, J. Funct. Anal., 113 (1993), 32-58.  doi: 10.1006/jfan.1993.1062.

[13]

S. Itô, Diffusion Equations, Transl. Math. Monogr., vol. 114, American Mathematical Society, Providence, RI, 1992.

[14]

W. Kryszewski and A. Szulkin, An infinite dimensional morse theorem with applications, Trans. Am. Math. Soc., 349 (1997), 3184-3234.  doi: 10.1090/S0002-9947-97-01963-6.

[15]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441-472. 

[16]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, 1971.

[17]

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. 

[18]

G. B. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.  doi: 10.1142/S0219199702000853.

[19]

G. B. Li and J. F. Yang, Asymptotically linear elliptic systems, Commun. Part. Diffe. Equ., 29 (2004), 925-954.  doi: 10.1081/PDE-120037337.

[20]

M. Nagasawa, Schrödinger Equations and Diffusion Theory, Birkhäuser, 1993.

[21]

A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.  doi: 10.1007/s00032-005-0047-8.

[22]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.

[23]

B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc., 7 (1982), 447-526.  doi: 10.1090/S0273-0979-1982-15041-8.

[24]

B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $\mathbb{R}^{N}$, Adv. Differential Equations, 5 (2000), 1445-1464. 

[25]

A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.  doi: 10.1016/j.jfa.2009.09.013.

[26]

E. A. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. PDE, 39 (2010), 1-33.  doi: 10.1007/s00526-009-0299-1.

[27]

E. A. Silva and G. F. Vieira, Quasilinear asymptotically periodic schrödinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935-2949.  doi: 10.1016/j.na.2009.11.037.

[28]

X. H. Tang, Non-Nehari manifold method for superlinear Schrödinger equation, Taiwan J. Math., 18 (2014), 1957-1979.  doi: 10.11650/tjm.18.2014.3541.

[29]

X. H. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58 (2015), 715-728.  doi: 10.1007/s11425-014-4957-1.

[30]

M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.

[31]

M. B. YangW. X. Chen and Y. H. Ding, Solutions of a class of Hamiltonian elliptic systems in $\mathbb{R}^{N}$, J. Math. Anal. Appl., 352 (2010), 338-349.  doi: 10.1016/j.jmaa.2009.07.052.

[32]

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.

[33]

F. K. ZhaoL. 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.

[34]

J. ZhangX. 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.

[35]

J. ZhangW. Zhang and X. Tang, Ground state solutions for Hamiltonian elliptic system with inverse square potential, Discrete Contin. Dyn. Syst., 37 (2017), 4565-4583.  doi: 10.3934/dcds.2017195.

[36]

J. ZhangX. 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.

[37]

J. ZhangX. Tang and W. Zhang, On semiclassical ground states for Hamiltonian elliptic system with critical growth, Topol. Meth. Nonl. Anal., 49 (2017), 245-272.  doi: 10.12775/tmna.2016.069.

[38]

W. ZhangJ ZhangZh ang and H. Mi, On fractional Schrödinger equation with periodic and asymptotically periodic conditions, Compu. Math. Appl., 74 (2017), 1321-1332.  doi: 10.1016/j.camwa.2017.06.017.

[39]

J. ZhangW. Zhang and X. H. Tang, Semiclassical limits of ground states for Hamiltonian elliptic system with gradient term, Nonlinear Anal. Real World Appl., 40 (2018), 377-402.  doi: 10.1016/j.nonrwa.2017.08.010.

[40]

J. ZhangW. Zhang and X. L. Xie, Existence and concentration of semiclassical solutions for Hamiltonian elliptic system, Comm. Pure Appl. Anal., 15 (2016), 599-622.  doi: 10.3934/cpaa.2016.15.599.

[41]

H. ZhangJ. X. Xu and F. B. Zhang, On a class of semilinear Schrödinger equations with indefinite linear part, J. Math. Anal. Appl., 414 (2014), 710-724.  doi: 10.1016/j.jmaa.2014.01.001.

show all references

References:
[1]

C. O. AlvesJ. M. do Ó and O. H. Miyagaki, On nonlinear perturbation of a periodic elliptic problem in $\mathbb{R}^2$ involving critical growth, Nonlinear Anal., 56 (2004), 781-791.  doi: 10.1016/j.na.2003.06.003.

[2]

S. Alama and Y. Y. Li, On "multibump" bound states for certain semilinear elliptic equations, Indiana Univ. Math. J., 41 (1992), 983-1026.  doi: 10.1512/iumj.1992.41.41052.

[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]

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.

[5]

V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math., 52 (1979), 241-273.  doi: 10.1007/BF01389883.

[6] Y. H. Ding, Variational Methods for Strongly Indefinite Problems, World Scientific Press, 2008.  doi: 10.1142/9789812709639.
[7]

D. G. De Figueiredo and Y. H. Ding, Strongly indefinite functions and multiple solutions of elliptic systems, Trans. Am. Math. Soc., 355 (2003), 2973-2989.  doi: 10.1090/S0002-9947-03-03257-4.

[8]

D. G. De Figueiredo and P. L. Felmer, On superquadiatic elliptic systems, Trans. Am. Math. Soc., 343 (1994), 97-116.  doi: 10.2307/2154523.

[9]

D. G. De Figueiredo, Semilinear elliptic systems: existence, multiplicity, symmetry of solutions, Handbook of Differential Equations Stationary Partial Differential Equations, 5, Elsevier, 2008, p.1–48. Chapter1. doi: 10.1016/S1874-5733(08)80008-3.

[10]

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.

[11]

M. J. Esteban and E. Séré, Stationary states of nonlinear Dirac equations: A variational approach, Comm. Math. Phys., 171 (1995), 323-350. 

[12]

J. Hulshof and R. C. A. M. De Vorst, Differential systems with strongly variational structure, J. Funct. Anal., 113 (1993), 32-58.  doi: 10.1006/jfan.1993.1062.

[13]

S. Itô, Diffusion Equations, Transl. Math. Monogr., vol. 114, American Mathematical Society, Providence, RI, 1992.

[14]

W. Kryszewski and A. Szulkin, An infinite dimensional morse theorem with applications, Trans. Am. Math. Soc., 349 (1997), 3184-3234.  doi: 10.1090/S0002-9947-97-01963-6.

[15]

W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441-472. 

[16]

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, 1971.

[17]

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. 

[18]

G. B. Li and A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776.  doi: 10.1142/S0219199702000853.

[19]

G. B. Li and J. F. Yang, Asymptotically linear elliptic systems, Commun. Part. Diffe. Equ., 29 (2004), 925-954.  doi: 10.1081/PDE-120037337.

[20]

M. Nagasawa, Schrödinger Equations and Diffusion Theory, Birkhäuser, 1993.

[21]

A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287.  doi: 10.1007/s00032-005-0047-8.

[22]

P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.  doi: 10.1007/BF00946631.

[23]

B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc., 7 (1982), 447-526.  doi: 10.1090/S0273-0979-1982-15041-8.

[24]

B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $\mathbb{R}^{N}$, Adv. Differential Equations, 5 (2000), 1445-1464. 

[25]

A. Szulkin and T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822.  doi: 10.1016/j.jfa.2009.09.013.

[26]

E. A. Silva and G. F. Vieira, Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var. PDE, 39 (2010), 1-33.  doi: 10.1007/s00526-009-0299-1.

[27]

E. A. Silva and G. F. Vieira, Quasilinear asymptotically periodic schrödinger equations with subcritical growth, Nonlinear Anal., 72 (2010), 2935-2949.  doi: 10.1016/j.na.2009.11.037.

[28]

X. H. Tang, Non-Nehari manifold method for superlinear Schrödinger equation, Taiwan J. Math., 18 (2014), 1957-1979.  doi: 10.11650/tjm.18.2014.3541.

[29]

X. H. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equations, Sci. China Math., 58 (2015), 715-728.  doi: 10.1007/s11425-014-4957-1.

[30]

M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1.

[31]

M. B. YangW. X. Chen and Y. H. Ding, Solutions of a class of Hamiltonian elliptic systems in $\mathbb{R}^{N}$, J. Math. Anal. Appl., 352 (2010), 338-349.  doi: 10.1016/j.jmaa.2009.07.052.

[32]

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.

[33]

F. K. ZhaoL. 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.

[34]

J. ZhangX. 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.

[35]

J. ZhangW. Zhang and X. Tang, Ground state solutions for Hamiltonian elliptic system with inverse square potential, Discrete Contin. Dyn. Syst., 37 (2017), 4565-4583.  doi: 10.3934/dcds.2017195.

[36]

J. ZhangX. 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.

[37]

J. ZhangX. Tang and W. Zhang, On semiclassical ground states for Hamiltonian elliptic system with critical growth, Topol. Meth. Nonl. Anal., 49 (2017), 245-272.  doi: 10.12775/tmna.2016.069.

[38]

W. ZhangJ ZhangZh ang and H. Mi, On fractional Schrödinger equation with periodic and asymptotically periodic conditions, Compu. Math. Appl., 74 (2017), 1321-1332.  doi: 10.1016/j.camwa.2017.06.017.

[39]

J. ZhangW. Zhang and X. H. Tang, Semiclassical limits of ground states for Hamiltonian elliptic system with gradient term, Nonlinear Anal. Real World Appl., 40 (2018), 377-402.  doi: 10.1016/j.nonrwa.2017.08.010.

[40]

J. ZhangW. Zhang and X. L. Xie, Existence and concentration of semiclassical solutions for Hamiltonian elliptic system, Comm. Pure Appl. Anal., 15 (2016), 599-622.  doi: 10.3934/cpaa.2016.15.599.

[41]

H. ZhangJ. X. Xu and F. B. Zhang, On a class of semilinear Schrödinger equations with indefinite linear part, J. Math. Anal. Appl., 414 (2014), 710-724.  doi: 10.1016/j.jmaa.2014.01.001.

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