# American Institute of Mathematical Sciences

August  2017, 37(8): 4565-4583. doi: 10.3934/dcds.2017195

## Ground state solutions for Hamiltonian elliptic system with inverse square potential

 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  October 2016 Revised  February 2017 Published  April 2017

Fund Project: This work was supported by the NNSF (Nos. 11601145,11571370,11471137), by the Natural Science Foundation of Hunan Province (Nos. 2017JJ3130,2017JJ3131), and by the Hunan University of Commerce Innovation Driven Project for Young Teacher (16QD008)

In this paper, we study the following Hamiltonian elliptic system with gradient term and inverse square potential
 $\left\{ \begin{array}{ll}-\Delta u +\vec{b}(x)\cdot \nabla u +V(x)u-\frac{\mu}{|x|^{2}}v=H_{v}(x,u,v)\\-\Delta v -\vec{b}(x)\cdot \nabla v +V(x)v-\frac{\mu}{|x|^{2}}u=H_{u}(x,u,v)\\\end{array} \right.$
for $x\in\mathbb{R}^{N}$, where $N\geq3$, $\mu\in\mathbb{R}$, and $V(x)$, $\vec{b}(x)$ and $H(x, u, v)$ are $1$-periodic in $x$. Under suitable conditions, we prove that the system possesses a ground state solution via variational methods for sufficiently small $\mu\geq0$. Moreover, we provide the comparison of the energy of ground state solutions for the case $\mu>0$ and $\mu=0$. Finally, we also give the convergence property of ground state solutions as $\mu\to0^+$.
Citation: Jian Zhang, Wen Zhang, Xianhua Tang. Ground state solutions for Hamiltonian elliptic system with inverse square potential. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4565-4583. doi: 10.3934/dcds.2017195
##### References:
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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 [31] 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 [32] 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 [33] 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 [34] 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 [35] 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 [36] 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 [37] F. K. Zhao, L. G. Zhao and Y. H. Ding, Multiple solution for a superlinear and periodic ellipic system on $\mathbb{R}^N$, Z. Angew. Math. Phys., 62 (2011), 495-511. doi: 10.1007/s00033-010-0105-0. Google Scholar

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##### References:
 [1] 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 [2] 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 [3] 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 [4] D. Cao and P. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential, J. Differential Equations, 205 (2004), 521-537. doi: 10.1016/j.jde.2004.03.005. Google Scholar [5] D. Cao and S. Peng, A note on the sign-changing solutions to elliptic problem with critical Sobolev and Hardy terms, J. Differential Equations, 193 (2003), 424-434. doi: 10.1016/S0022-0396(03)00118-9. Google Scholar [6] D. Cao and S. Peng, A global compactness result for singular elliptic problems involving critical Sobolev exponent, Proc. Amer. Math. Soc., 131 (2003), 1857-1866. doi: 10.1090/S0002-9939-02-06729-1. Google Scholar [7] D. Cao and S. Yan, Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential, Calc. Var., 38 (2010), 471-501. doi: 10.1007/s00526-009-0295-5. Google Scholar [8] Z. Chen and W. Zou, On an elliptic problem with critical exponent and Hardy potential, J. Differential Equations, 252 (2012), 969-987. doi: 10.1016/j.jde.2011.09.042. 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), 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., 33 (1998), 211-234. doi: 10.1016/S0362-546X(97)00548-8. Google Scholar [11] Y. Deng, L. Jin and S. Peng, Solutions of Schrödinger equations with inverse square potential and critical nonlinearity, J. Differential Equations, 253 (2012), 1376-1398. doi: 10.1016/j.jde.2012.05.009. Google Scholar [12] Y. H. Ding, Variational Methods for Strongly Indefinite Problems, World Scientific Press, 2007. doi: 10.1142/9789812709639. Google Scholar [13] V. Felli, On the existence of ground state solutions to nonlinear Schrödinger equations with multisingular inverse-square anisotropic potentials, J. Anal. Math., 108 (2009), 189-217. doi: 10.1007/s11854-009-0023-2. Google Scholar [14] V. Felli and S. Terracini, Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity, Comm. Partial Differential Equations, 31 (2006), 469-495. doi: 10.1080/03605300500394439. Google Scholar [15] V. Felli, E. Marchini and S. Terracini, On Schrödinger operators with multipolar inversesquare potentials, J. Funct. Anal., 250 (2007), 265-316. doi: 10.1016/j.jfa.2006.10.019. Google Scholar [16] Q. Guo and J. Mederski, Ground states of nonlinear Schrödinger equations with sum of periodic and inverse-square potentials, J. Differential Equations, 260 (2016), 4180-4202. doi: 10.1016/j.jde.2015.11.006. Google Scholar [17] W. Kryszewki and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441-472. Google Scholar [18] G. Li and J. Yang, Asymptotically linear elliptic systems, Commun. Part. Diffe. Equ., 29 (2004), 925-954. doi: 10.1081/PDE-120037337. Google Scholar [19] 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 [20] 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. doi: 10.1016/S0294-1449(16)30422-X. 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] D. Ruiz and M. Willem, Elliptic problems with critical exponents and Hardy potentials, J. Differential Equations, 190 (2003), 524-538. doi: 10.1016/S0022-0396(02)00178-X. Google Scholar [23] B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in RN, Adv. Differential Equations, 5 (2000), 1445-1464. Google Scholar [24] D. Smets, Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities, Trans. Amer. Math. Soc., 357 (2005), 2909-2938. doi: 10.1090/S0002-9947-04-03769-9. 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] 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 [27] M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1. Google Scholar [28] M. B. Yang, W. X. Chen and Y. H. Ding, Solutions of a class of Hamiltonian elliptic systems in ${{\rm{\mathbb{R}}}^N}$, J. Math. Anal. Appl., 352 (2010), 338-349. doi: 10.1016/j.jmaa.2009.07.052. Google Scholar [29] J. Zhang, X. H. Tang and W. Zhang, Ground states for diffusion system with periodic and asymptotically periodic nonlinearity, Comput. Math. Appl., 71 (2016), 633-641. doi: 10.1016/j.camwa.2015.12.031. Google Scholar [30] 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 [31] 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 [32] 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 [33] 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 [34] 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 [35] 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 [36] 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 [37] F. K. Zhao, L. G. Zhao and Y. H. Ding, Multiple solution for a superlinear and periodic ellipic system on $\mathbb{R}^N$, Z. Angew. Math. Phys., 62 (2011), 495-511. doi: 10.1007/s00033-010-0105-0. Google Scholar
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