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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 & Applied Analysis, 2019, 18 (5) : 2433-2455. doi: 10.3934/cpaa.2019110
##### References:
 [1] C. O. Alves, J. 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. Google Scholar [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. 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] 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 [5] V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math., 52 (1979), 241-273. doi: 10.1007/BF01389883. Google Scholar [6] Y. H. Ding, Variational Methods for Strongly Indefinite Problems, World Scientific Press, 2008. doi: 10.1142/9789812709639. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. 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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 [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. Google Scholar [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. Google Scholar [20] M. Nagasawa, Schrödinger Equations and Diffusion Theory, Birkhäuser, 1993. Google Scholar [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. Google Scholar [22] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631. Google Scholar [23] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc., 7 (1982), 447-526. doi: 10.1090/S0273-0979-1982-15041-8. Google Scholar [24] B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $\mathbb{R}^{N}$, Adv. Differential Equations, 5 (2000), 1445-1464. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [30] M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1. Google Scholar [31] M. B. Yang, W. 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. Google Scholar [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. Google Scholar [33] 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 [34] J. Zhang, X. 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 [35] J. Zhang, W. 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. Google Scholar [36] J. Zhang, X. 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 [37] J. Zhang, X. 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. Google Scholar [38] W. Zhang, J Zhang, Zh 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. Google Scholar [39] J. Zhang, W. 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. Google Scholar [40] J. Zhang, W. 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. Google Scholar [41] H. Zhang, J. 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. Google Scholar

show all references

##### References:
 [1] C. O. Alves, J. 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. Google Scholar [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. 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] 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 [5] V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math., 52 (1979), 241-273. doi: 10.1007/BF01389883. Google Scholar [6] Y. H. Ding, Variational Methods for Strongly Indefinite Problems, World Scientific Press, 2008. doi: 10.1142/9789812709639. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [11] M. J. Esteban and E. Séré, Stationary states of nonlinear Dirac equations: A variational approach, Comm. Math. Phys., 171 (1995), 323-350. Google Scholar [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. Google Scholar [13] S. Itô, Diffusion Equations, Transl. Math. Monogr., vol. 114, American Mathematical Society, Providence, RI, 1992. Google Scholar [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. Google Scholar [15] W. Kryszewski and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441-472. Google Scholar [16] J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, 1971. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [20] M. Nagasawa, Schrödinger Equations and Diffusion Theory, Birkhäuser, 1993. Google Scholar [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. Google Scholar [22] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291. doi: 10.1007/BF00946631. Google Scholar [23] B. Simon, Schrödinger semigroups, Bull. Amer. Math. Soc., 7 (1982), 447-526. doi: 10.1090/S0273-0979-1982-15041-8. Google Scholar [24] B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in $\mathbb{R}^{N}$, Adv. Differential Equations, 5 (2000), 1445-1464. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [30] M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1. Google Scholar [31] M. B. Yang, W. 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. Google Scholar [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. Google Scholar [33] 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 [34] J. Zhang, X. 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 [35] J. Zhang, W. 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. Google Scholar [36] J. Zhang, X. 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 [37] J. Zhang, X. 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. Google Scholar [38] W. Zhang, J Zhang, Zh 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. Google Scholar [39] J. Zhang, W. 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. Google Scholar [40] J. Zhang, W. 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. Google Scholar [41] H. Zhang, J. 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. Google Scholar
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