# American Institute of Mathematical Sciences

September 2019, 18(5): 2663-2677. doi: 10.3934/cpaa.2019118

## Qualitative properties of stationary solutions of the NLS on the Hyperbolic space without and with external potentials

 Department of Mathematics and Statistics, University of Ottawa, 150 Louis Pasteur Private (ON) K1N 7N5 Canada

* Corresponding author

Received  August 2018 Revised  October 2018 Published  April 2019

In this paper, we prove some qualitative properties of stationary solutions of the NLS on the Hyperbolic space. First, we prove a variational characterization of the ground state and give a complete characterization of the spectrum of the linearized operator around the ground state. Then we prove some rigidity theorems and necessary conditions for the existence of solutions in weighted spaces. Finally, we add a slowly varying potential to the homogeneous equation and prove the existence of non-trivial solutions concentrating on the critical points of a reduced functional. The results are the natural counterparts of the corresponding theorems on the Euclidean space. We produce also the natural virial identity on the Hyperbolic space for the complete evolution, which however requires the introduction of a weighted energy, which is not conserved and so does not lead directly to finite time blow-up as in the Euclidean case.

Citation: Alessandro Selvitella. Qualitative properties of stationary solutions of the NLS on the Hyperbolic space without and with external potentials. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2663-2677. doi: 10.3934/cpaa.2019118
##### References:
 [1] A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbb{R}^n$, 1$^{st}$ edition, Birkhäuser Verlag, Basel, 2006. [2] A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, 1$^{st}$ edition, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511618260. [3] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381. [4] V. Banica, The nonlinear Schrödinger equation on hyperbolic space, Comm. Partial Differential Equations, 32 (2007), 1643-1677. doi: 10.1080/03605300600854332. [5] V. Banica, R. Carles and T. Duyckaerts, On scattering for NLS: from Euclidean to hyperbolic space, Disc. Contin. Dyn. Syst., 24 (2009), 1113-1127. doi: 10.3934/dcds.2009.24.1113. [6] V. Banica and T. Duyckaerts, Global existence, scattering and blow-up for the focusing NLS on the hyperbolic space, Dyn. Partial Differ. Equ., 12 (2015), 53-96. doi: 10.4310/DPDE.2015.v12.n1.a4. [7] V. Banica, R. Carles and G. Staffilani, Scattering theory for radial nonlinear Schrödinger equations on hyperbolic space, Geom. Funct. Anal., 18 (2008), 367-399. doi: 10.1007/s00039-008-0663-x. [8] M. Bhakta and K. Sandeep, Poincaré-Sobolev equations in the hyperbolic space, Calc. Var. Partial Differential Equations, 44 (2012), 247-269. doi: 10.1007/s00526-011-0433-8. [9] H. Christianson and J. Marzuola, Existence and stability of solitons for the nonlinear Schrödinger equation on hyperbolic space, Nonlinearity, 23 (2010), 89-106. doi: 10.1088/0951-7715/23/1/005. [10] R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, Journal of Mathematical Physics, 18 (1977), 1794-1797. doi: 10.1063/1.523491. [11] G. Mancini and K. Sandeep, On a semilinear elliptic equation in $\mathbb{H}^n$, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (2008), 635-671. [12] R. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30. [13] A. A. Pankov, Invariant semilinear elliptic equations on a manifold of constant negative curvature, Funktsional. Anal. i Prilozhen., 26 (1992), 82-84. doi: 10.1007/BF01075639. [14] K. Sandeep and D. Ganguly, Nondegeneracy of positive solutions of semilinear elliptic problems in the hyperbolic space, Communications in Contemporary Mathematics, 5 (2015), 13pp. doi: 10.1142/S0219199714500199. [15] A. Selvitella, ODE based proofs of uniqueness and nondegeneracy of the ground state of the NLS on $\mathbb{H}^d$ with positive imaginary phase, submitted. [16] T. Tao, Shock Waves and Reaction-Diffusion Equations, 2$^{nd}$ edition, CBMS Regional Conference Series in Mathematics, 106 Washington, DC, 2006. [17] Z. Wang, Uniqueness of radial solutions of semilinear elliptic equations on hyperbolic space, Nonlinear Analysis, 104 (2014), 109-119. doi: 10.1016/j.na.2014.03.015.

show all references

##### References:
 [1] A. Ambrosetti and A. Malchiodi, Perturbation Methods and Semilinear Elliptic Problems on $\mathbb{R}^n$, 1$^{st}$ edition, Birkhäuser Verlag, Basel, 2006. [2] A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, 1$^{st}$ edition, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511618260. [3] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis, 14 (1973), 349-381. [4] V. Banica, The nonlinear Schrödinger equation on hyperbolic space, Comm. Partial Differential Equations, 32 (2007), 1643-1677. doi: 10.1080/03605300600854332. [5] V. Banica, R. Carles and T. Duyckaerts, On scattering for NLS: from Euclidean to hyperbolic space, Disc. Contin. Dyn. Syst., 24 (2009), 1113-1127. doi: 10.3934/dcds.2009.24.1113. [6] V. Banica and T. Duyckaerts, Global existence, scattering and blow-up for the focusing NLS on the hyperbolic space, Dyn. Partial Differ. Equ., 12 (2015), 53-96. doi: 10.4310/DPDE.2015.v12.n1.a4. [7] V. Banica, R. Carles and G. Staffilani, Scattering theory for radial nonlinear Schrödinger equations on hyperbolic space, Geom. Funct. Anal., 18 (2008), 367-399. doi: 10.1007/s00039-008-0663-x. [8] M. Bhakta and K. Sandeep, Poincaré-Sobolev equations in the hyperbolic space, Calc. Var. Partial Differential Equations, 44 (2012), 247-269. doi: 10.1007/s00526-011-0433-8. [9] H. Christianson and J. Marzuola, Existence and stability of solitons for the nonlinear Schrödinger equation on hyperbolic space, Nonlinearity, 23 (2010), 89-106. doi: 10.1088/0951-7715/23/1/005. [10] R. T. Glassey, On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, Journal of Mathematical Physics, 18 (1977), 1794-1797. doi: 10.1063/1.523491. [11] G. Mancini and K. Sandeep, On a semilinear elliptic equation in $\mathbb{H}^n$, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (2008), 635-671. [12] R. Palais, The principle of symmetric criticality, Comm. Math. Phys., 69 (1979), 19-30. [13] A. A. Pankov, Invariant semilinear elliptic equations on a manifold of constant negative curvature, Funktsional. Anal. i Prilozhen., 26 (1992), 82-84. doi: 10.1007/BF01075639. [14] K. Sandeep and D. Ganguly, Nondegeneracy of positive solutions of semilinear elliptic problems in the hyperbolic space, Communications in Contemporary Mathematics, 5 (2015), 13pp. doi: 10.1142/S0219199714500199. [15] A. Selvitella, ODE based proofs of uniqueness and nondegeneracy of the ground state of the NLS on $\mathbb{H}^d$ with positive imaginary phase, submitted. [16] T. Tao, Shock Waves and Reaction-Diffusion Equations, 2$^{nd}$ edition, CBMS Regional Conference Series in Mathematics, 106 Washington, DC, 2006. [17] Z. Wang, Uniqueness of radial solutions of semilinear elliptic equations on hyperbolic space, Nonlinear Analysis, 104 (2014), 109-119. doi: 10.1016/j.na.2014.03.015.
 [1] Xiaoming He, Marco Squassina, Wenming Zou. The Nehari manifold for fractional systems involving critical nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1285-1308. doi: 10.3934/cpaa.2016.15.1285 [2] Yukihiro Seki. A remark on blow-up at space infinity. Conference Publications, 2009, 2009 (Special) : 691-696. doi: 10.3934/proc.2009.2009.691 [3] Qingfang Wang. The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2261-2281. doi: 10.3934/cpaa.2018108 [4] Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119 [5] Caisheng Chen, Qing Yuan. Existence of solution to $p-$Kirchhoff type problem in $\mathbb{R}^N$ via Nehari manifold. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2289-2303. doi: 10.3934/cpaa.2014.13.2289 [6] A. Pankov. Gap solitons in periodic discrete nonlinear Schrödinger equations II: A generalized Nehari manifold approach. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 419-430. doi: 10.3934/dcds.2007.19.419 [7] Yi-hang Hao, Xian-Gao Liu. The existence and blow-up criterion of liquid crystals system in critical Besov space. Communications on Pure & Applied Analysis, 2014, 13 (1) : 225-236. doi: 10.3934/cpaa.2014.13.225 [8] Shota Sato. Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1225-1237. doi: 10.3934/cpaa.2011.10.1225 [9] Yuta Wakasugi. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831 [10] Baoquan Yuan, Xiao Li. Blow-up criteria of smooth solutions to the three-dimensional micropolar fluid equations in Besov space. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2167-2179. doi: 10.3934/dcdss.2016090 [11] Valeria Banica, Rémi Carles, Thomas Duyckaerts. On scattering for NLS: From Euclidean to hyperbolic space. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1113-1127. doi: 10.3934/dcds.2009.24.1113 [12] Baiyu Liu, Li Ma. Blow up threshold for a parabolic type equation involving space integral and variational structure. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2169-2183. doi: 10.3934/cpaa.2015.14.2169 [13] Giovanna Citti, Maria Manfredini, Alessandro Sarti. Finite difference approximation of the Mumford and Shah functional in a contact manifold of the Heisenberg space. Communications on Pure & Applied Analysis, 2010, 9 (4) : 905-927. doi: 10.3934/cpaa.2010.9.905 [14] Agust Sverrir Egilsson. On embedding the $1:1:2$ resonance space in a Poisson manifold. Electronic Research Announcements, 1995, 1: 48-56. [15] Hongyu Ye. Positive high energy solution for Kirchhoff equation in $\mathbb{R}^{3}$ with superlinear nonlinearities via Nehari-Pohožaev manifold. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3857-3877. doi: 10.3934/dcds.2015.35.3857 [16] C. Y. Chan. Recent advances in quenching and blow-up of solutions. Conference Publications, 2001, 2001 (Special) : 88-95. doi: 10.3934/proc.2001.2001.88 [17] Marina Chugunova, Chiu-Yen Kao, Sarun Seepun. On the Benilov-Vynnycky blow-up problem. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1443-1460. doi: 10.3934/dcdsb.2015.20.1443 [18] Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1 [19] Marek Fila, Hiroshi Matano. Connecting equilibria by blow-up solutions. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 155-164. doi: 10.3934/dcds.2000.6.155 [20] Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399

2017 Impact Factor: 0.884