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January  2021, 20(1): 121-143. doi: 10.3934/cpaa.2020260

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

Noriyoshi Fukaya

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

Received  April 2020 Revised  August 2020 Published  January 2021 Early access  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.

Keywords: Uniqueness, nondegeneracy, ground state, nonlinear Schrödinger equation, inverse-power potential, standing wave, instability.
Mathematics Subject Classification: Primary: 35A02, 35Q55; Secondary: 35J61.
Citation: Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure and Applied Analysis, 2021, 20 (1) : 121-143. 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.

[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.

[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.

[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.

[5]

S. M. Chang, S. Gustafson, K. 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.

[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.

[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.

[8]

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

[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. 

[10]

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

[11]

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

[12]

M. Grillakis, J. 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.

[13]

M. Grillakis, J. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[24]

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

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[5]

S. M. Chang, S. Gustafson, K. 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.

[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.

[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.

[8]

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

[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. 

[10]

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

[11]

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

[12]

M. Grillakis, J. 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.

[13]

M. Grillakis, J. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[24]

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

[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.

[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.

[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.

[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.

[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.

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