# 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 and Continuous Dynamical Systems, 2017, 37 (8) : 4565-4583. doi: 10.3934/dcds.2017195
##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [12] Y. H. Ding, Variational Methods for Strongly Indefinite Problems, World Scientific Press, 2007. doi: 10.1142/9789812709639. [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. [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. [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. [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. [17] W. Kryszewki and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441-472. [18] G. Li and J. Yang, Asymptotically linear elliptic systems, Commun. Part. Diffe. Equ., 29 (2004), 925-954.  doi: 10.1081/PDE-120037337. [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. [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. [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] 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. [23] B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in RN, Adv. Differential Equations, 5 (2000), 1445-1464. [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. [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] 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. [27] M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [12] Y. H. Ding, Variational Methods for Strongly Indefinite Problems, World Scientific Press, 2007. doi: 10.1142/9789812709639. [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. [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. [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. [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. [17] W. Kryszewki and A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441-472. [18] G. Li and J. Yang, Asymptotically linear elliptic systems, Commun. Part. Diffe. Equ., 29 (2004), 925-954.  doi: 10.1081/PDE-120037337. [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. [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. [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] 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. [23] B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in RN, Adv. Differential Equations, 5 (2000), 1445-1464. [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. [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] 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. [27] M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996. doi: 10.1007/978-1-4612-4146-1. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
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