# American Institute of Mathematical Sciences

May  2019, 18(3): 1547-1565. doi: 10.3934/cpaa.2019074

## Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential

 1 Department of Mathematics, Huaihua College, Huaihua, Hunan 418008, China 2 School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

* Corresponding author

Received  July 2017 Revised  December 2017 Published  November 2018

We consider the following semilinear Schrödinger equation with inverse square potential
 \begin{array}{l}\left\{ \begin{align} & -\vartriangle u+(V(x)-\frac{\mu }{|x{{|}^{2}}}u=f(x,u),\ \ \ \ \ x\in {{\mathbb{R}}^{N}}, \\ & u\in {{H}^{1}}({{\mathbb{R}}^{N}}), \\ \end{align} \right.\end{array}
where $N≥ 3$, $f$ is asymptotically linear, $V$ is 1-periodic in each of $x_1, ..., x_N$ and $\sup[σ(-\triangle +V)\cap (-∞, 0)]＜0＜{\rm{inf}}[σ(-\triangle +V)\cap (0, ∞)]$. Under some mild assumptions on $V$ and $f$, we prove the existence and asymptotical behavior of ground state solutions of Nehari-Pankov type to the above problem.
Citation: Xiaoyan Lin, Yubo He, Xianhua Tang. Existence and asymptotic behavior of ground state solutions for asymptotically linear Schrödinger equation with inverse square potential. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1547-1565. doi: 10.3934/cpaa.2019074
##### References:
 [1] 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. [2] J. Chabrowski, A. Szulkin and M. Willem, Schrödinger equation with multiparticle potential and critical nonlinearity, Topol. Methods Nonlinear Anal., 34 (2009), 201-211. doi: 10.12775/TMNA.2009.038. [3] S.T. Chen and X.H. Tang, Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete Contin Dyn Syst-A, 38 (2018), 2333-2348. doi: 10.3934/dcds.2018096. [4] V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on ${\mathbb{R}}^n$, Comm. Pure Appl. Math., 45 (1992), 1217-1269. doi: 10.1002/cpa.3160451002. [5] Y. B. Deng, L. Y. Jin and S. J. 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. [6] Y. H. Ding, Variational Methods for Strongly Indefinite Problems World Scientific, Singapore, 2007. doi: 10.1142/9789812709639. [7] Y. H. Ding and C. Lee, Multiple solutions of Schröinger equations with indefinite linear part and super or asymptotically linear terms, J. Differential Equations, 222 (2006), 137-163. doi: 10.1016/j.jde.2005.03.011. [8] D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987. [9] Y. Egorov and V. Kondratiev, On Spectral Theory of Elliptic Operators, Birkhäuser, Basel, 1996. doi: 10.1007/978-3-0348-9029-8. [10] 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. [11] V. Felli, E. Marchini and S. Terracini, On Schrödinger operators with multisingular inverse-square anisotropic potentials, J. Funct. Anal., 250 (2007), 265-316. doi: 10.1016/j.jfa.2006.10.019. [12] V. Felli, E. Marchini and S. Terracini, On Schrödinger operators with multisingular inverse-square anisotropic potentials, Indiana Univ. Math. J., 58 (2009), 617-676. doi: 10.1512/iumj.2009.58.3471. [13] V. Felli and A. Primo, Classification of local asymptotics for solutions to heat equations with inverse-square potentials, Disc. Contin. Dyn. Syst., 31 (2011), 65-107. doi: 10.3934/dcds.2011.31.65. [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] Q. Guo and J. Mederski, Ground states of nonlinear Schrödinger equations with sum of periodic and inverse-square potential, J. Differential Equations, 260 (2016), 4180-4202. doi: 10.1016/j.jde.2015.11.006. [16] J. Jeanjean, On the existence of bounded Palais-Smale sequence and application to a Landesman-Lazer-type problem on ${\mathbb{R}}^N$, Proc. Roc. Soc. Edinberg, 129 (1999), 787-809. doi: 10.1017/S0308210500013147. [17] X. Y. Lin and X. H. Tang, An asymptotically periodic and asymptotically linear Schrödinger equation with indefinite linear part, Comput. Math. Applic., 70 (2015), 726-736. doi: 10.1016/j.camwa.2015.06.013. [18] W. Kryszewski and A. Szulkin, Generalized linking theoremwith an application to a semilinear Schrödinger equation, Adv. Differ. Equ., 3 (1998), 441-472. [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] S. Liu, On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), 1-9. doi: 10.1007/s00526-011-0447-2. [21] J. Mederski, Solutions to a nonlinear Schrödinger equation with periodic potential and zero on the boundary of the spectrum, Topol. Methods Nonlinear Anal., 46 (2015), 755-771. [22] J. Mederski, Ground states of a system of nonlinear Schrödinger equations with periodic potentials, Comm. Partial Differential Equations, 41 (2016), 1426-1440. doi: 10.1080/03605302.2016.1209520. [23] 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. [24] 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. [25] 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. [26] 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. [27] A. Szulkin and W. M. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems, J. Funct. Anal., 187 (2001), 25-41. doi: 10.1006/jfan.2001.3798. [28] X. H. Tang, New conditions on nonlinearity for a periodic Schrödinger equation having zero as spectrum, J. Math. Anal. Appl., 413 (2014), 392-410. doi: 10.1016/j.jmaa.2013.11.062. [29] X. H. Tang, Ground state solutions for superlinear Schrödinger equation, Advance Nonlinear Studies, 14 (2014), 361-373. doi: 10.1515/ans-2014-0208. [30] X. H. Tang, Non-Nehari manifold method for asymptotically linear Schrödinger equation, J. Aust. Math. Soc., 98 (2015), 104-116. doi: 10.1017/S144678871400041X. [31] X. H. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equation, Sci. China Math., 58 (2015), 715-728. doi: 10.1007/s11425-014-4957-1. [32] X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potential, Disc. Contin. Dyn. Syst., 37 (2017), 4973-5002. doi: 10.3934/dcds.2017214. [33] X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohožaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations, 56 (2017), 110-134. doi: 10.1007/s00526-017-1214-9. [34] S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 2 (1996), 241-264. [35] L. Wei, X. Y. Cheng and Z. S. Feng, Exact behavior of positive solutions to elliptic equations with multi-singular inverse square potentials, Disc. Contin. Dyn. Syst., 36 (2016), 7169-7189. doi: 10.3934/dcds.2016112. [36] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1. [37] L. Zhang, X. H. Tang and Y. Chen, Infinitely many solutions for a class of perturbed elliptic equations with nonlocal operators, Commun. Pur. Appl. Anal., 16 (2017), 823-842. doi: 10.3934/cpaa.2017039. [38] J. Zhang, W. Zhang and X. H. Tang, Ground state solutions for Hamiltonian elliptic system with inverse square potential, Disc. Contin. Dyn. Syst., 37 (2017), 4565-4583. doi: 10.3934/dcds.2017195.

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##### References:
 [1] 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. [2] J. Chabrowski, A. Szulkin and M. Willem, Schrödinger equation with multiparticle potential and critical nonlinearity, Topol. Methods Nonlinear Anal., 34 (2009), 201-211. doi: 10.12775/TMNA.2009.038. [3] S.T. Chen and X.H. Tang, Improved results for Klein-Gordon-Maxwell systems with general nonlinearity, Discrete Contin Dyn Syst-A, 38 (2018), 2333-2348. doi: 10.3934/dcds.2018096. [4] V. Coti Zelati and P. H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on ${\mathbb{R}}^n$, Comm. Pure Appl. Math., 45 (1992), 1217-1269. doi: 10.1002/cpa.3160451002. [5] Y. B. Deng, L. Y. Jin and S. J. 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. [6] Y. H. Ding, Variational Methods for Strongly Indefinite Problems World Scientific, Singapore, 2007. doi: 10.1142/9789812709639. [7] Y. H. Ding and C. Lee, Multiple solutions of Schröinger equations with indefinite linear part and super or asymptotically linear terms, J. Differential Equations, 222 (2006), 137-163. doi: 10.1016/j.jde.2005.03.011. [8] D. E. Edmunds and W. D. Evans, Spectral Theory and Differential Operators, Clarendon Press, Oxford, 1987. [9] Y. Egorov and V. Kondratiev, On Spectral Theory of Elliptic Operators, Birkhäuser, Basel, 1996. doi: 10.1007/978-3-0348-9029-8. [10] 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. [11] V. Felli, E. Marchini and S. Terracini, On Schrödinger operators with multisingular inverse-square anisotropic potentials, J. Funct. Anal., 250 (2007), 265-316. doi: 10.1016/j.jfa.2006.10.019. [12] V. Felli, E. Marchini and S. Terracini, On Schrödinger operators with multisingular inverse-square anisotropic potentials, Indiana Univ. Math. J., 58 (2009), 617-676. doi: 10.1512/iumj.2009.58.3471. [13] V. Felli and A. Primo, Classification of local asymptotics for solutions to heat equations with inverse-square potentials, Disc. Contin. Dyn. Syst., 31 (2011), 65-107. doi: 10.3934/dcds.2011.31.65. [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] Q. Guo and J. Mederski, Ground states of nonlinear Schrödinger equations with sum of periodic and inverse-square potential, J. Differential Equations, 260 (2016), 4180-4202. doi: 10.1016/j.jde.2015.11.006. [16] J. Jeanjean, On the existence of bounded Palais-Smale sequence and application to a Landesman-Lazer-type problem on ${\mathbb{R}}^N$, Proc. Roc. Soc. Edinberg, 129 (1999), 787-809. doi: 10.1017/S0308210500013147. [17] X. Y. Lin and X. H. Tang, An asymptotically periodic and asymptotically linear Schrödinger equation with indefinite linear part, Comput. Math. Applic., 70 (2015), 726-736. doi: 10.1016/j.camwa.2015.06.013. [18] W. Kryszewski and A. Szulkin, Generalized linking theoremwith an application to a semilinear Schrödinger equation, Adv. Differ. Equ., 3 (1998), 441-472. [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] S. Liu, On superlinear Schrödinger equations with periodic potential, Calc. Var. Partial Differential Equations, 45 (2012), 1-9. doi: 10.1007/s00526-011-0447-2. [21] J. Mederski, Solutions to a nonlinear Schrödinger equation with periodic potential and zero on the boundary of the spectrum, Topol. Methods Nonlinear Anal., 46 (2015), 755-771. [22] J. Mederski, Ground states of a system of nonlinear Schrödinger equations with periodic potentials, Comm. Partial Differential Equations, 41 (2016), 1426-1440. doi: 10.1080/03605302.2016.1209520. [23] 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. [24] 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. [25] 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. [26] 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. [27] A. Szulkin and W. M. Zou, Homoclinic orbits for asymptotically linear Hamiltonian systems, J. Funct. Anal., 187 (2001), 25-41. doi: 10.1006/jfan.2001.3798. [28] X. H. Tang, New conditions on nonlinearity for a periodic Schrödinger equation having zero as spectrum, J. Math. Anal. Appl., 413 (2014), 392-410. doi: 10.1016/j.jmaa.2013.11.062. [29] X. H. Tang, Ground state solutions for superlinear Schrödinger equation, Advance Nonlinear Studies, 14 (2014), 361-373. doi: 10.1515/ans-2014-0208. [30] X. H. Tang, Non-Nehari manifold method for asymptotically linear Schrödinger equation, J. Aust. Math. Soc., 98 (2015), 104-116. doi: 10.1017/S144678871400041X. [31] X. H. Tang, Non-Nehari manifold method for asymptotically periodic Schrödinger equation, Sci. China Math., 58 (2015), 715-728. doi: 10.1007/s11425-014-4957-1. [32] X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potential, Disc. Contin. Dyn. Syst., 37 (2017), 4973-5002. doi: 10.3934/dcds.2017214. [33] X. H. Tang and S. T. Chen, Ground state solutions of Nehari-Pohožaev type for Kirchhoff-type problems with general potentials, Calc. Var. Partial Differential Equations, 56 (2017), 110-134. doi: 10.1007/s00526-017-1214-9. [34] S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 2 (1996), 241-264. [35] L. Wei, X. Y. Cheng and Z. S. Feng, Exact behavior of positive solutions to elliptic equations with multi-singular inverse square potentials, Disc. Contin. Dyn. Syst., 36 (2016), 7169-7189. doi: 10.3934/dcds.2016112. [36] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1. [37] L. Zhang, X. H. Tang and Y. Chen, Infinitely many solutions for a class of perturbed elliptic equations with nonlocal operators, Commun. Pur. Appl. Anal., 16 (2017), 823-842. doi: 10.3934/cpaa.2017039. [38] J. Zhang, W. Zhang and X. H. Tang, Ground state solutions for Hamiltonian elliptic system with inverse square potential, Disc. Contin. Dyn. Syst., 37 (2017), 4565-4583. doi: 10.3934/dcds.2017195.
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