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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 
$ \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.$ 
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), 348376. 
[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), 5167. 
[3] 
T. Bartsch, Y. H. Ding, Deformation theorems on nonmetrizable vector spaces and applications to critical point theory, Math. Nach., 279 (2006), 12671288. 
[4] 
D. Cao, P. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential, J. Differential Equations, 205 (2004), 521537. 
[5] 
D. Cao, S. Peng, A note on the signchanging solutions to elliptic problem with critical Sobolev and Hardy terms, J. Differential Equations, 193 (2003), 424434. 
[6] 
D. Cao, S. Peng, A global compactness result for singular elliptic problems involving critical Sobolev exponent, Proc. Amer. Math. Soc., 131 (2003), 18571866. 
[7] 
D. Cao, S. Yan, Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential, Calc. Var., 38 (2010), 471501. 
[8] 
Z. Chen, W. Zou, On an elliptic problem with critical exponent and Hardy potential, J. Differential Equations, 252 (2012), 969987. 
[9] 
D. G. De Figueiredo, Semilinear elliptic systems: existence, multiplicity, symmetry of solutions, Handbook of Differential Equations Stationary Partial Differential Equations, 5, Elsevier, (2008), 148. Chapter1. 
[10] 
D. G. De Figueiredo, J. Yang, Decay, Symmetry and existence of solutions of semilinear elliptic systems, Nonlinear. Anal., 33 (1998), 211234. 
[11] 
Y. Deng, L. Jin, S. Peng, Solutions of Schrödinger equations with inverse square potential and critical nonlinearity, J. Differential Equations, 253 (2012), 13761398. 
[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 inversesquare anisotropic potentials, J. Anal. Math., 108 (2009), 189217. 
[14] 
V. Felli, S. Terracini, Elliptic equations with multisingular inversesquare potentials and critical nonlinearity, Comm. Partial Differential Equations, 31 (2006), 469495. 
[15] 
V. Felli, E. Marchini, S. Terracini, On Schrödinger operators with multipolar inversesquare potentials, J. Funct. Anal., 250 (2007), 265316. 
[16] 
Q. Guo, J. Mederski, Ground states of nonlinear Schrödinger equations with sum of periodic and inversesquare potentials, J. Differential Equations, 260 (2016), 41804202. 
[17] 
W. Kryszewki, A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441472. 
[18] 
G. Li, J. Yang, Asymptotically linear elliptic systems, Commun. Part. Diffe. Equ., 29 (2004), 925954. 
[19] 
G. B. Li, A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763776. 
[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), 223283. 
[21] 
A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259287. 
[22] 
D. Ruiz, M. Willem, Elliptic problems with critical exponents and Hardy potentials, J. Differential Equations, 190 (2003), 524538. 
[23] 
B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in RN, Adv. Differential Equations, 5 (2000), 14451464. 
[24] 
D. Smets, Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities, Trans. Amer. Math. Soc., 357 (2005), 29092938. 
[25] 
A. Szulkin, T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 38023822. 
[26] 
X. H. Tang, NonNehari manifold method for superlinear Schrödinger equation, Taiwan J. Math., 18 (2014), 19571979. 
[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), 338349. 
[29] 
J. Zhang, X. H. Tang, W. Zhang, Ground states for diffusion system with periodic and asymptotically periodic nonlinearity, Comput. Math. Appl., 71 (2016), 633641. 
[30] 
J. Zhang, X. H. Tang, W. Zhang, Groundstate solutions for superquadratic Hamiltonian elliptic systems with gradient terms, Nonlinear Anal., 95 (2014), 110. 
[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), 357371. 
[32] 
J. Zhang, X. H. Tang, W. Zhang, On semiclassical ground state solutions for Hamiltonian elliptic systems, Appl. Anal., 94 (2015), 13801396. 
[33] 
J. Zhang, W. Zhang, X. L. Xie, Existence and concentration of semiclassical solutions for Hamiltonian elliptic system, Comm. Pure Appl. Anal., 15 (2016), 599622. 
[34] 
F. K. Zhao, Y. H. Ding, On Hamiltonian elliptic systems with periodic or nonperiodic potentials, J. Differential Equations, 249 (2010), 29642985. 
[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), 7791. 
[36] 
F. K. Zhao, L. G. Zhao, Y. H. Ding, Multiple solutions for asympototically linear elliptic systems, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 673688. 
[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), 495511. 
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), 348376. 
[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), 5167. 
[3] 
T. Bartsch, Y. H. Ding, Deformation theorems on nonmetrizable vector spaces and applications to critical point theory, Math. Nach., 279 (2006), 12671288. 
[4] 
D. Cao, P. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential, J. Differential Equations, 205 (2004), 521537. 
[5] 
D. Cao, S. Peng, A note on the signchanging solutions to elliptic problem with critical Sobolev and Hardy terms, J. Differential Equations, 193 (2003), 424434. 
[6] 
D. Cao, S. Peng, A global compactness result for singular elliptic problems involving critical Sobolev exponent, Proc. Amer. Math. Soc., 131 (2003), 18571866. 
[7] 
D. Cao, S. Yan, Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential, Calc. Var., 38 (2010), 471501. 
[8] 
Z. Chen, W. Zou, On an elliptic problem with critical exponent and Hardy potential, J. Differential Equations, 252 (2012), 969987. 
[9] 
D. G. De Figueiredo, Semilinear elliptic systems: existence, multiplicity, symmetry of solutions, Handbook of Differential Equations Stationary Partial Differential Equations, 5, Elsevier, (2008), 148. Chapter1. 
[10] 
D. G. De Figueiredo, J. Yang, Decay, Symmetry and existence of solutions of semilinear elliptic systems, Nonlinear. Anal., 33 (1998), 211234. 
[11] 
Y. Deng, L. Jin, S. Peng, Solutions of Schrödinger equations with inverse square potential and critical nonlinearity, J. Differential Equations, 253 (2012), 13761398. 
[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 inversesquare anisotropic potentials, J. Anal. Math., 108 (2009), 189217. 
[14] 
V. Felli, S. Terracini, Elliptic equations with multisingular inversesquare potentials and critical nonlinearity, Comm. Partial Differential Equations, 31 (2006), 469495. 
[15] 
V. Felli, E. Marchini, S. Terracini, On Schrödinger operators with multipolar inversesquare potentials, J. Funct. Anal., 250 (2007), 265316. 
[16] 
Q. Guo, J. Mederski, Ground states of nonlinear Schrödinger equations with sum of periodic and inversesquare potentials, J. Differential Equations, 260 (2016), 41804202. 
[17] 
W. Kryszewki, A. Szulkin, Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441472. 
[18] 
G. Li, J. Yang, Asymptotically linear elliptic systems, Commun. Part. Diffe. Equ., 29 (2004), 925954. 
[19] 
G. B. Li, A. Szulkin, An asymptotically periodic Schrödinger equation with indefinite linear part, Commun. Contemp. Math., 4 (2002), 763776. 
[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), 223283. 
[21] 
A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259287. 
[22] 
D. Ruiz, M. Willem, Elliptic problems with critical exponents and Hardy potentials, J. Differential Equations, 190 (2003), 524538. 
[23] 
B. Sirakov, On the existence of solutions of Hamiltonian elliptic systems in RN, Adv. Differential Equations, 5 (2000), 14451464. 
[24] 
D. Smets, Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities, Trans. Amer. Math. Soc., 357 (2005), 29092938. 
[25] 
A. Szulkin, T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 38023822. 
[26] 
X. H. Tang, NonNehari manifold method for superlinear Schrödinger equation, Taiwan J. Math., 18 (2014), 19571979. 
[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), 338349. 
[29] 
J. Zhang, X. H. Tang, W. Zhang, Ground states for diffusion system with periodic and asymptotically periodic nonlinearity, Comput. Math. Appl., 71 (2016), 633641. 
[30] 
J. Zhang, X. H. Tang, W. Zhang, Groundstate solutions for superquadratic Hamiltonian elliptic systems with gradient terms, Nonlinear Anal., 95 (2014), 110. 
[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), 357371. 
[32] 
J. Zhang, X. H. Tang, W. Zhang, On semiclassical ground state solutions for Hamiltonian elliptic systems, Appl. Anal., 94 (2015), 13801396. 
[33] 
J. Zhang, W. Zhang, X. L. Xie, Existence and concentration of semiclassical solutions for Hamiltonian elliptic system, Comm. Pure Appl. Anal., 15 (2016), 599622. 
[34] 
F. K. Zhao, Y. H. Ding, On Hamiltonian elliptic systems with periodic or nonperiodic potentials, J. Differential Equations, 249 (2010), 29642985. 
[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), 7791. 
[36] 
F. K. Zhao, L. G. Zhao, Y. H. Ding, Multiple solutions for asympototically linear elliptic systems, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 673688. 
[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), 495511. 
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