# 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 and 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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