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 and 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. BanicaR. 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. BanicaR. 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. BanicaR. 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. BanicaR. 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.

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