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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 & 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.  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. 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.  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. 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.  Google Scholar

[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.  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. 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.  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. 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.  Google Scholar

[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.  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

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