# American Institue of Mathematical Sciences

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

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:
 [1] A. I. Ávila, J. Yang, On the existence and shape of least energy solutions for some elliptic systems, J. Differential Equations, 191 (2003), 348-376. [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, Y. H. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Math. Nach., 279 (2006), 1267-1288. [4] D. Cao, P. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential, J. Differential Equations, 205 (2004), 521-537. [5] D. Cao, 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. [6] D. Cao, S. Peng, A global compactness result for singular elliptic problems involving critical Sobolev exponent, Proc. Amer. Math. Soc., 131 (2003), 1857-1866. [7] D. Cao, S. Yan, Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential, Calc. Var., 38 (2010), 471-501. [8] Z. Chen, W. Zou, On an elliptic problem with critical exponent and Hardy potential, J. Differential Equations, 252 (2012), 969-987. [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. [10] D. G. De Figueiredo, J. Yang, Decay, Symmetry and existence of solutions of semilinear elliptic systems, Nonlinear. Anal., 33 (1998), 211-234. [11] Y. Deng, L. Jin, S. Peng, Solutions of Schrödinger equations with inverse square potential and critical nonlinearity, J. Differential Equations, 253 (2012), 1376-1398. [12] Y. H. Ding, Variational Methods for Strongly Indefinite Problems, World Scientific Press, 2007. [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. [14] V. Felli, S. Terracini, Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity, Comm. Partial Differential Equations, 31 (2006), 469-495. [15] V. Felli, E. Marchini, S. Terracini, On Schrödinger operators with multipolar inversesquare potentials, J. Funct. Anal., 250 (2007), 265-316. [16] Q. Guo, J. Mederski, Ground states of nonlinear Schrödinger equations with sum of periodic and inverse-square potentials, J. Differential Equations, 260 (2016), 4180-4202. [17] W. Kryszewki, A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441-472. [18] G. Li, J. Yang, Asymptotically linear elliptic systems, Commun. Part. Diffe. Equ., 29 (2004), 925-954. [19] G. B. Li, A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776. [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. [21] A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287. [22] D. Ruiz, M. Willem, Elliptic problems with critical exponents and Hardy potentials, J. Differential Equations, 190 (2003), 524-538. [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. [25] A. Szulkin, T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822. [26] X. H. Tang, Non-Nehari manifold method for superlinear Schrödinger equation, Taiwan J. Math., 18 (2014), 1957-1979. [27] M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996. [28] M. B. Yang, W. X. Chen, Y. H. Ding, Solutions of a class of Hamiltonian elliptic systems in RN, J. Math. Anal. Appl., 352 (2010), 338-349. [29] J. Zhang, X. H. Tang, W. Zhang, Ground states for diffusion system with periodic and asymptotically periodic nonlinearity, Comput. Math. Appl., 71 (2016), 633-641. [30] J. Zhang, X. H. Tang, W. Zhang, Ground-state solutions for superquadratic Hamiltonian elliptic systems with gradient terms, Nonlinear Anal., 95 (2014), 1-10. [31] J. Zhang, X. H. Tang, W. Zhang, Semiclassical solutions for a class of Schrödinger system with magnetic potentials, J. Math. Anal. Appl., 414 (2014), 357-371. [32] J. Zhang, X. H. Tang, W. Zhang, On semiclassical ground state solutions for Hamiltonian elliptic systems, Appl. Anal., 94 (2015), 1380-1396. [33] J. Zhang, W. Zhang, X. L. Xie, Existence and concentration of semiclassical solutions for Hamiltonian elliptic system, Comm. Pure Appl. Anal., 15 (2016), 599-622. [34] F. K. Zhao, Y. H. Ding, On Hamiltonian elliptic systems with periodic or non-periodic potentials, J. Differential Equations, 249 (2010), 2964-2985. [35] F. K. Zhao, L. G. Zhao, Y. H. Ding, Infinitly mang solutions for asymptotically linear periodic Hamiltonian system, ESAIM: Control, Optim. Calc. Vari., 16 (2010), 77-91. [36] F. K. Zhao, L. G. Zhao, Y. H. Ding, Multiple solutions for asympototically linear elliptic systems, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 673-688. [37] F. K. Zhao, L. G. Zhao, Y. H. Ding, Multiple solution for a superlinear and periodic ellipic system on $\mathbb{R}^N$, Z. Angew. Math. Phys., 62 (2011), 495-511.

show all references

##### References:
 [1] A. I. Ávila, J. Yang, On the existence and shape of least energy solutions for some elliptic systems, J. Differential Equations, 191 (2003), 348-376. [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, Y. H. Ding, Deformation theorems on non-metrizable vector spaces and applications to critical point theory, Math. Nach., 279 (2006), 1267-1288. [4] D. Cao, P. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential, J. Differential Equations, 205 (2004), 521-537. [5] D. Cao, 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. [6] D. Cao, S. Peng, A global compactness result for singular elliptic problems involving critical Sobolev exponent, Proc. Amer. Math. Soc., 131 (2003), 1857-1866. [7] D. Cao, S. Yan, Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential, Calc. Var., 38 (2010), 471-501. [8] Z. Chen, W. Zou, On an elliptic problem with critical exponent and Hardy potential, J. Differential Equations, 252 (2012), 969-987. [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. [10] D. G. De Figueiredo, J. Yang, Decay, Symmetry and existence of solutions of semilinear elliptic systems, Nonlinear. Anal., 33 (1998), 211-234. [11] Y. Deng, L. Jin, S. Peng, Solutions of Schrödinger equations with inverse square potential and critical nonlinearity, J. Differential Equations, 253 (2012), 1376-1398. [12] Y. H. Ding, Variational Methods for Strongly Indefinite Problems, World Scientific Press, 2007. [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. [14] V. Felli, S. Terracini, Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity, Comm. Partial Differential Equations, 31 (2006), 469-495. [15] V. Felli, E. Marchini, S. Terracini, On Schrödinger operators with multipolar inversesquare potentials, J. Funct. Anal., 250 (2007), 265-316. [16] Q. Guo, J. Mederski, Ground states of nonlinear Schrödinger equations with sum of periodic and inverse-square potentials, J. Differential Equations, 260 (2016), 4180-4202. [17] W. Kryszewki, A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441-472. [18] G. Li, J. Yang, Asymptotically linear elliptic systems, Commun. Part. Diffe. Equ., 29 (2004), 925-954. [19] G. B. Li, A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763-776. [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. [21] A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259-287. [22] D. Ruiz, M. Willem, Elliptic problems with critical exponents and Hardy potentials, J. Differential Equations, 190 (2003), 524-538. [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. [25] A. Szulkin, T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802-3822. [26] X. H. Tang, Non-Nehari manifold method for superlinear Schrödinger equation, Taiwan J. Math., 18 (2014), 1957-1979. [27] M. Willem, Minimax Theorems, Birkhäuser, Berlin, 1996. [28] M. B. Yang, W. X. Chen, Y. H. Ding, Solutions of a class of Hamiltonian elliptic systems in RN, J. Math. Anal. Appl., 352 (2010), 338-349. [29] J. Zhang, X. H. Tang, W. Zhang, Ground states for diffusion system with periodic and asymptotically periodic nonlinearity, Comput. Math. Appl., 71 (2016), 633-641. [30] J. Zhang, X. H. Tang, W. Zhang, Ground-state solutions for superquadratic Hamiltonian elliptic systems with gradient terms, Nonlinear Anal., 95 (2014), 1-10. [31] J. Zhang, X. H. Tang, W. Zhang, Semiclassical solutions for a class of Schrödinger system with magnetic potentials, J. Math. Anal. Appl., 414 (2014), 357-371. [32] J. Zhang, X. H. Tang, W. Zhang, On semiclassical ground state solutions for Hamiltonian elliptic systems, Appl. Anal., 94 (2015), 1380-1396. [33] J. Zhang, W. Zhang, X. L. Xie, Existence and concentration of semiclassical solutions for Hamiltonian elliptic system, Comm. Pure Appl. Anal., 15 (2016), 599-622. [34] F. K. Zhao, Y. H. Ding, On Hamiltonian elliptic systems with periodic or non-periodic potentials, J. Differential Equations, 249 (2010), 2964-2985. [35] F. K. Zhao, L. G. Zhao, Y. H. Ding, Infinitly mang solutions for asymptotically linear periodic Hamiltonian system, ESAIM: Control, Optim. Calc. Vari., 16 (2010), 77-91. [36] F. K. Zhao, L. G. Zhao, Y. H. Ding, Multiple solutions for asympototically linear elliptic systems, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 673-688. [37] F. K. Zhao, L. G. Zhao, Y. H. Ding, Multiple solution for a superlinear and periodic ellipic system on $\mathbb{R}^N$, Z. Angew. Math. Phys., 62 (2011), 495-511.
 [1] Yaoping Chen, Jianqing Chen. Existence of multiple positive weak solutions and estimates for extremal values for a class of concave-convex elliptic problems with an inverse-square potential. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1531-1552. doi: 10.3934/cpaa.2017073 [2] Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179 [3] Scipio Cuccagna, Masaya Maeda. On weak interaction between a ground state and a trapping potential. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3343-3376. doi: 10.3934/dcds.2015.35.3343 [4] Gisèle Ruiz Goldstein, Jerome A. Goldstein, Abdelaziz Rhandi. Kolmogorov equations perturbed by an inverse-square potential. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 623-630. doi: 10.3934/dcdss.2011.4.623 [5] Emmanuel Hebey, Jérôme Vétois. Multiple solutions for critical elliptic systems in potential form. Communications on Pure & Applied Analysis, 2008, 7 (3) : 715-741. doi: 10.3934/cpaa.2008.7.715 [6] Rumei Zhang, Jin Chen, Fukun Zhao. Multiple solutions for superlinear elliptic systems of Hamiltonian type. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1249-1262. doi: 10.3934/dcds.2011.30.1249 [7] Lei Wei, Xiyou Cheng, Zhaosheng Feng. Exact behavior of positive solutions to elliptic equations with multi-singular inverse square potentials. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7169-7189. doi: 10.3934/dcds.2016112 [8] Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. $L^p$ Estimates for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 427-442. doi: 10.3934/dcds.2003.9.427 [9] Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. Dispersive estimate for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1387-1400. doi: 10.3934/dcds.2003.9.1387 [10] Hengguang Li, Jeffrey S. Ovall. A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1377-1391. doi: 10.3934/dcdsb.2015.20.1377 [11] Rowan Killip, Changxing Miao, Monica Visan, Junyong Zhang, Jiqiang Zheng. The energy-critical NLS with inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3831-3866. doi: 10.3934/dcds.2017162 [12] Leszek Gasiński, Nikolaos S. Papageorgiou. Multiplicity of solutions for Neumann problems with an indefinite and unbounded potential. Communications on Pure & Applied Analysis, 2013, 12 (5) : 1985-1999. doi: 10.3934/cpaa.2013.12.1985 [13] Dung Le. On the regular set of BMO weak solutions to $p$-Laplacian strongly coupled nonregular elliptic systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3245-3265. doi: 10.3934/dcdsb.2014.19.3245 [14] Xianjin Chen, Jianxin Zhou. A local min-orthogonal method for multiple solutions of strongly coupled elliptic systems. Conference Publications, 2009, 2009 (Special) : 151-160. doi: 10.3934/proc.2009.2009.151 [15] D. Motreanu, V. V. Motreanu, Nikolaos S. Papageorgiou. Nonautonomous resonant periodic systems with indefinite linear part and a nonsmooth potential. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1401-1414. doi: 10.3934/cpaa.2011.10.1401 [16] B. Buffoni, F. Giannoni. Brake periodic orbits of prescribed Hamiltonian for indefinite Lagrangian systems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 217-222. doi: 10.3934/dcds.1995.1.217 [17] 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 [18] C. Cortázar, Marta García-Huidobro. On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian. Communications on Pure & Applied Analysis, 2006, 5 (4) : 813-826. doi: 10.3934/cpaa.2006.5.813 [19] Norihisa Ikoma. Existence of ground state solutions to the nonlinear Kirchhoff type equations with potentials. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 943-966. doi: 10.3934/dcds.2015.35.943 [20] C. Cortázar, Marta García-Huidobro. On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian. Communications on Pure & Applied Analysis, 2006, 5 (1) : 71-84. doi: 10.3934/cpaa.2006.5.71

2016 Impact Factor: 1.099