• Previous Article
    Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional
  • CPAA Home
  • This Issue
  • Next Article
    Isomorphism between one-Dimensional and multidimensional finite difference operators
doi: 10.3934/cpaa.2020260

Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential

Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan

Received  April 2020 Revised  August 2020 Published  October 2020

Fund Project: This work was supported by Grant-in-Aid for JSPS Fellows 18J11090 and JSPS KAKENHI Grant Number 20K14349

We study uniqueness and nondegeneracy of ground states for stationary nonlinear Schrödinger equations with a focusing power-type nonlinearity and an attractive inverse-power potential. We refine the results of Shioji and Watanabe (2016) and apply it to prove the uniqueness and nondegeneracy of ground states for our equations. We also discuss the orbital instability of ground state-standing waves.

Citation: Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2020260
References:
[1]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.  Google Scholar

[2]

J. Byeon and Y. Oshita, Uniqueness of standing waves for nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 975-987.  doi: 10.1017/S0308210507000236.  Google Scholar

[3]

T. Cazenave, An introduction to nonlinear Schrödinger equations, vol. 22 of Textos de Métodos Matemáticos, Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, 1989. Google Scholar

[4]

T. Cazenave, Semilinear Schrödinger equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003 doi: 10.1090/cln/010.  Google Scholar

[5]

S. M. ChangS. GustafsonK. Nakanishi and T. P. Tsai, Spectra of linearized operators for NLS solitary waves, SIAM J. Math. Anal., 39 (2007/08), 1070-1111.  doi: 10.1137/050648389.  Google Scholar

[6]

C. V. Coffman, Uniqueness of the ground state solution for $\Delta u-u+u^{3} = 0$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95.  doi: 10.1007/BF00250684.  Google Scholar

[7]

A. Comech and D. Pelinovsky, Purely nonlinear instability of standing waves with minimal energy, Commun. Pure Appl. Math., 56 (2003), 1565-1607.  doi: 10.1002/cpa.10104.  Google Scholar

[8]

V. D. Dinh, On nonlinear Schrödinger equations with attractive inverse-power potentials, arXiv: 1903.04636. Google Scholar

[9]

N. Fukaya and M. Ohta, Strong instability of standing waves for nonlinear Schrödinger equations with attractive inverse power potential, Osaka J. Math., 56 (2019), 713-726.   Google Scholar

[10]

R. Fukuizumi and M. Ohta, Instability of standing waves for nonlinear Schrödinger equations with potentials, Differ. Integral Equ., 16 (2003), 691-706.   Google Scholar

[11]

R. Fukuizumi and M. Ohta, Stability of standing waves for nonlinear Schrödinger equations with potentials, Differ. Integral Equ., 16 (2003), 111-128.   Google Scholar

[12]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. Ⅰ, J. Funct. Anal., 74 (1987), 160-197.  doi: 10.1016/0022-1236(87)90044-9.  Google Scholar

[13]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. Ⅱ, J. Funct. Anal., 94 (1990), 308-348.  doi: 10.1016/0022-1236(90)90016-E.  Google Scholar

[14]

Y. Kabeya and K. Tanaka, Uniqueness of positive radial solutions of semilinear elliptic equations in ${\bf R}^N$ and Séré's non-degeneracy condition, Commun. Partial Differ. Equ., 24 (1999), 563-598.  doi: 10.1080/03605309908821434.  Google Scholar

[15]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in ${\bf{R}}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar

[16]

X. Li and J. Zhao, Orbital stability of standing waves for Schrödinger type equations with slowly decaying linear potential, Comput. Math. Appl., 79 (2020), 303-316.  doi: 10.1016/j.camwa.2019.06.030.  Google Scholar

[17]

E. H. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[18]

M. Maeda, Stability of bound states of Hamiltonian PDEs in the degenerate cases, J. Funct. Anal., 263 (2012), 511-528.  doi: 10.1016/j.jfa.2012.04.006.  Google Scholar

[19]

K. McLeod and J. Serrin, Uniqueness of positive radial solutions of $\Delta u+f(u) = 0$ in ${\bf{R}}^n$, Arch. Rational Mech. Anal., 99 (1987), 115-145.  doi: 10.1007/BF00275874.  Google Scholar

[20]

W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281.  doi: 10.1215/S0012-7094-93-07004-4.  Google Scholar

[21]

M. Ohta, Instability of bound states for abstract nonlinear Schrödinger equations, J. Funct. Anal., 261 (2011), 90-110.  doi: 10.1016/j.jfa.2011.03.010.  Google Scholar

[22]

L. A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in ${\bf{R}}^n$, Arch. Rational Mech. Anal., 81 (1983), 181-197.  doi: 10.1007/BF00250651.  Google Scholar

[23]

J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations, Commun. Math. Phys., 91 (1983), 313-327.   Google Scholar

[24]

J. Shatah and W. Strauss, Instability of nonlinear bound states, Commun. Math. Phys., 100 (1985), 173-190.   Google Scholar

[25]

N. Shioji and K. Watanabe, A generalized Pohožaev identity and uniqueness of positive radial solutions of $\Delta u+g(r)u+h(r)u^p = 0$, J. Differ. Equ., 255 (2013), 4448-4475.  doi: 10.1016/j.jde.2013.08.017.  Google Scholar

[26]

N. Shioji and K. Watanabe, Uniqueness and nondegeneracy of positive radial solutions of $ {\rm div} (\rho\nabla u)+\rho(-gu+hu^p) = 0$, Calc. Var. Partial Differ. Equ., 55 (2016), Art. 32, 42. doi: 10.1007/s00526-016-0970-2.  Google Scholar

[27]

M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491.  doi: 10.1137/0516034.  Google Scholar

[28]

M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Commun. Pure Appl. Math., 39 (1986), 51-67.  doi: 10.1002/cpa.3160390103.  Google Scholar

[29]

E. Yanagida, Uniqueness of positive radial solutions of $\Delta u+g(r)u+h(r)u^p=0$ in ${\bf{R}}^n$, Arch. Rational Mech. Anal., 115 (1991), 257-274.  doi: 10.1007/BF00380770.  Google Scholar

show all references

References:
[1]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.  doi: 10.1007/BF00250555.  Google Scholar

[2]

J. Byeon and Y. Oshita, Uniqueness of standing waves for nonlinear Schrödinger equations, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 975-987.  doi: 10.1017/S0308210507000236.  Google Scholar

[3]

T. Cazenave, An introduction to nonlinear Schrödinger equations, vol. 22 of Textos de Métodos Matemáticos, Instituto de Matemática, Universidade Federal do Rio de Janeiro, Rio de Janeiro, 1989. Google Scholar

[4]

T. Cazenave, Semilinear Schrödinger equations, vol. 10 of Courant Lecture Notes in Mathematics, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003 doi: 10.1090/cln/010.  Google Scholar

[5]

S. M. ChangS. GustafsonK. Nakanishi and T. P. Tsai, Spectra of linearized operators for NLS solitary waves, SIAM J. Math. Anal., 39 (2007/08), 1070-1111.  doi: 10.1137/050648389.  Google Scholar

[6]

C. V. Coffman, Uniqueness of the ground state solution for $\Delta u-u+u^{3} = 0$ and a variational characterization of other solutions, Arch. Rational Mech. Anal., 46 (1972), 81-95.  doi: 10.1007/BF00250684.  Google Scholar

[7]

A. Comech and D. Pelinovsky, Purely nonlinear instability of standing waves with minimal energy, Commun. Pure Appl. Math., 56 (2003), 1565-1607.  doi: 10.1002/cpa.10104.  Google Scholar

[8]

V. D. Dinh, On nonlinear Schrödinger equations with attractive inverse-power potentials, arXiv: 1903.04636. Google Scholar

[9]

N. Fukaya and M. Ohta, Strong instability of standing waves for nonlinear Schrödinger equations with attractive inverse power potential, Osaka J. Math., 56 (2019), 713-726.   Google Scholar

[10]

R. Fukuizumi and M. Ohta, Instability of standing waves for nonlinear Schrödinger equations with potentials, Differ. Integral Equ., 16 (2003), 691-706.   Google Scholar

[11]

R. Fukuizumi and M. Ohta, Stability of standing waves for nonlinear Schrödinger equations with potentials, Differ. Integral Equ., 16 (2003), 111-128.   Google Scholar

[12]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. Ⅰ, J. Funct. Anal., 74 (1987), 160-197.  doi: 10.1016/0022-1236(87)90044-9.  Google Scholar

[13]

M. GrillakisJ. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. Ⅱ, J. Funct. Anal., 94 (1990), 308-348.  doi: 10.1016/0022-1236(90)90016-E.  Google Scholar

[14]

Y. Kabeya and K. Tanaka, Uniqueness of positive radial solutions of semilinear elliptic equations in ${\bf R}^N$ and Séré's non-degeneracy condition, Commun. Partial Differ. Equ., 24 (1999), 563-598.  doi: 10.1080/03605309908821434.  Google Scholar

[15]

M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p = 0$ in ${\bf{R}}^n$, Arch. Rational Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.  Google Scholar

[16]

X. Li and J. Zhao, Orbital stability of standing waves for Schrödinger type equations with slowly decaying linear potential, Comput. Math. Appl., 79 (2020), 303-316.  doi: 10.1016/j.camwa.2019.06.030.  Google Scholar

[17]

E. H. Lieb and M. Loss, Analysis, vol. 14 of Graduate Studies in Mathematics, 2nd edition, American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/014.  Google Scholar

[18]

M. Maeda, Stability of bound states of Hamiltonian PDEs in the degenerate cases, J. Funct. Anal., 263 (2012), 511-528.  doi: 10.1016/j.jfa.2012.04.006.  Google Scholar

[19]

K. McLeod and J. Serrin, Uniqueness of positive radial solutions of $\Delta u+f(u) = 0$ in ${\bf{R}}^n$, Arch. Rational Mech. Anal., 99 (1987), 115-145.  doi: 10.1007/BF00275874.  Google Scholar

[20]

W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281.  doi: 10.1215/S0012-7094-93-07004-4.  Google Scholar

[21]

M. Ohta, Instability of bound states for abstract nonlinear Schrödinger equations, J. Funct. Anal., 261 (2011), 90-110.  doi: 10.1016/j.jfa.2011.03.010.  Google Scholar

[22]

L. A. Peletier and J. Serrin, Uniqueness of positive solutions of semilinear equations in ${\bf{R}}^n$, Arch. Rational Mech. Anal., 81 (1983), 181-197.  doi: 10.1007/BF00250651.  Google Scholar

[23]

J. Shatah, Stable standing waves of nonlinear Klein-Gordon equations, Commun. Math. Phys., 91 (1983), 313-327.   Google Scholar

[24]

J. Shatah and W. Strauss, Instability of nonlinear bound states, Commun. Math. Phys., 100 (1985), 173-190.   Google Scholar

[25]

N. Shioji and K. Watanabe, A generalized Pohožaev identity and uniqueness of positive radial solutions of $\Delta u+g(r)u+h(r)u^p = 0$, J. Differ. Equ., 255 (2013), 4448-4475.  doi: 10.1016/j.jde.2013.08.017.  Google Scholar

[26]

N. Shioji and K. Watanabe, Uniqueness and nondegeneracy of positive radial solutions of $ {\rm div} (\rho\nabla u)+\rho(-gu+hu^p) = 0$, Calc. Var. Partial Differ. Equ., 55 (2016), Art. 32, 42. doi: 10.1007/s00526-016-0970-2.  Google Scholar

[27]

M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal., 16 (1985), 472-491.  doi: 10.1137/0516034.  Google Scholar

[28]

M. I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Commun. Pure Appl. Math., 39 (1986), 51-67.  doi: 10.1002/cpa.3160390103.  Google Scholar

[29]

E. Yanagida, Uniqueness of positive radial solutions of $\Delta u+g(r)u+h(r)u^p=0$ in ${\bf{R}}^n$, Arch. Rational Mech. Anal., 115 (1991), 257-274.  doi: 10.1007/BF00380770.  Google Scholar

[1]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[2]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[3]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020376

[4]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

[5]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020264

[6]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[7]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456

[8]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[9]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[10]

Xiyou Cheng, Zhitao Zhang. Structure of positive solutions to a class of Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020461

[11]

Shiqi Ma. On recent progress of single-realization recoveries of random Schrödinger systems. Electronic Research Archive, , () : -. doi: 10.3934/era.2020121

[12]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[13]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

[14]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[15]

Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466

[16]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[17]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[18]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103

[19]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076

[20]

Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang. Duality between range and no-response tests and its application for inverse problems. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020072

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (18)
  • HTML views (45)
  • Cited by (0)

Other articles
by authors

[Back to Top]