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
.
DOI: 10.1016/S0252-9602(18)30803-8
[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