A NOTE ON THE UNIQUENESS AND THE NON-DEGENERACY OF POSITIVE RADIAL SOLUTIONS FOR SEMILINEAR ELLIPTIC PROBLEMS AND ITS APPLICATION

doi:10.1016/S0252-9602(18)30803-8

Abstract

Abstract:

In this paper, we are concerned with the uniqueness and the non-degeneracy of positive radial solutions for a class of semilinear elliptic equations. Using detailed ODE analysis, we extend previous results to cases where nonlinear terms may have sublinear growth. As an application, we obtain the uniqueness and the non-degeneracy of ground states for modified Schrödinger equations.

Key words: positive radial solution, uniqueness, non-degeneracy, shooting method

Shinji ADACHI, Masataka SHIBATA, Tatsuya WATANABE. A NOTE ON THE UNIQUENESS AND THE NON-DEGENERACY OF POSITIVE RADIAL SOLUTIONS FOR SEMILINEAR ELLIPTIC PROBLEMS AND ITS APPLICATION[J].Acta mathematica scientia,Series B, 2018, 38(4): 1121-1142.

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References 30

[1]	Adachi S, Watanabe T. Uniqueness of the ground state solutions of quasilinear Schrödinger equations. Nonlinear Anal, 2012, 75:819-833
[2]	Adachi S, Watanabe T. Asymptotic uniqueness of ground states for a class of quasilinear Schrödinger equations with H1-supercritical exponent. J Differential Equations, 2016, 260:3086-3118
[3]	Adachi S, Shibata M, Watanabe T. Global uniqueness results for ground states for a class of quasilinear elliptic equations. Kodai Math J, 2017, 40:117-142
[4]	Bates P, Shi J. Existence and instability of spike layer solutions to singular perturbation problems. J Funct Anal, 2002, 196:429-482
[5]	Berestycki H, Gallouët T, Kavian O. Equations de champs scalaires euclidens non linéaires dans le plan. C R Acad Paris Sér I Math, 1984, 297:307-310
[6]	Berestycki H, Lions P L. Nonlinear scalar fields equations, I. Existence of a ground state. Arch Ration Mech Anal, 1983, 82:313-345
[7]	Brizhik L, Eremko A, Piette B, Zahkrzewski W J. Static solutions of a D-dimensional modified nonlinear Schrödinger equation. Nonlinearity, 2003, 161481-1497
[8]	Byeon J, Jeanjean L, Maris M. Symmetry and monotonicity of least energy solutions. Calc Var Partial Differential Equations, 2009, 36:481-492
[9]	Chen J, Li Y, Wang Z Q. Stability of standing waves for a class of quasilinear Schrödinger equations. European J Appl Math, 2012, 23:611-633
[10]	Coffman C V. Uniqueness of the ground state solution for △u-u+u3=0 and a variational characterization of other solutions. Arch Ration Mech Anal, 1972, 46:81-95
[11]	Colin M, Jeanjean L, Squassina M. Stability and instability results for standing waves of quasi-linear Schrödinger equations. Nonlinearity, 2010, 23:1353-1385
[12]	Colin M, Ohta M. Instability of ground states for a quasilinear Schrödinger equation. Differential Integral Equations, 2014, 27:613-624
[13]	Cortázar C, Elgueta M, Felmer P. Uniqueness of positive solutions of △u + f(u)=0 in RN, N ≥ 3. Arch Ration Mech Anal, 1998, 142:127-141
[14]	Floer A, Weinstein A. Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J Funct Anal, 1986, 69:397-408
[15]	Gidas B, Ni W M, Nirenberg L. Symmetry of positive solutions of nonlinear elliptic equations in RN. Adv Math Suppl Stud, 1981, 7A:369-402
[16]	Gladiali F, Squassina M. Uniqueness of ground states for a class of quasi-linear elliptic equations. Adv Nonlinear Anal, 2012, 1:159-179
[17]	Hirata J, Ikoma N, Tanaka K. Nonlinear scalar field equations in RN:mountain pass and symmetric mountain pass approaches. Topol Methods Nonlinear Anal, 2010, 35:253-276
[18]	Korman P. A global approach to ground state solutions. Electron J Differential Equations, 2008, 122:1-13
[19]	Kurihara S. Large-amplitude quasi-solitons in superfluid films. J Phys Soc Japan, 1981, 50:3262-3267
[20]	Kwong M K. Uniqueness of positive solutions of △u -u + up=0 in RN. Arch Ration Mech Anal, 1989, 105:243-266
[21]	Mariş M. Existence of nonstationary bubbles in higher dimensions. J Math Pures Appl, 2002, 81:1207-1239
[22]	Mcleod K. Uniqueness of positive radial solutions of △u+f(u)=0 in RN, Ⅱ. Trans Amer Math Soc, 1993, 339:495-505
[23]	Mcleod K, Serrin J. Uniqueness of positive radial solutions of △u=f(u)=0 in RN. Arch Ration Mech Anal, 1987, 99:115-145
[24]	Ni W M, Takagi I. Locating the peaks of least-energy solutions to a semilinear Neumann problem. Duke Math J, 1993, 70:247-281
[25]	Ouyang T, Shi J. Exact multiplicity of positive solutions for a class of semilinear problems:Ⅱ. J Differential Equations, 1999, 158:94-151
[26]	Peletier L A, Serrin J. Uniqueness of positive solutions of semilinear equations in Rn. Arch Ration Mech Anal, 1983, 81:181-197
[27]	Peletier L A, Serrin J. Uniqueness of non-negative solutions of semilinear equations in Rn. J Differential Equations, 1986, 61:380-397
[28]	Selvitella A. Nondegeneracy of the ground state for quasilinear Schrödinger equations. Calc Var Partial Differential Equations, 2015, 53:349-364
[29]	Serrin J, Tang M. Uniqueness of ground states for quasilinear elliptic equations. Indiana Univ Math J, 2000, 49:897-923
[30]	Stuart C. Lectures on the orbital stability of standing waves and applications to the nonlinear Schrödinger equation. Milan J Math, 2008, 76:329-399