MR2332079
Munemitsu HIROSE and Masahito OHTA
Munemitsu HIROSE and Masahito OHTA
Uniqueness of Positive Solutions to Scalar Field Equations with Harmonic Potential
Funkcialaj Ekvacioj. Serio Internacia
50
2007
67--100
http://fe.math.kobe-u.ac.jp/FE/FullPapers/50-1/50_67.pdf
http://www.ams.org/mathscinet-getitem?mr=MR2332079
Our purpose is to show the uniqueness of positive solutions for a nonlinear elliptic equation with a harmonic potential in some energy space. This problem arises in the study on the stability of standing waves for a nonlinear Schrödinger equation, and in [3], the existence of positive solutions to the ellipitc equation has been shown by the standard variational method. Moreover, in [5] we have derived the uniqueness under $n \geq 3$, where $n$ denotes the space dimension. In this paper, we give the proof of the uniqueness for $n=1$ and $n=2$.
Nonlinear elliptic equation, Harmonic potential, Uniqueness of solutions, Radial solutions.
34A12, 35J60.
50-67
2007
Uniqueness of Positive Solutions to Scalar Field Equations with Harmonic Potential
Munemitsu HIROSE and Masahito OHTA
Munemitsu HIROSE and Masahito OHTA
1
Cazenave, T.
An introduction to nonlinear Schrödinger equations
Textos de Métods Matemáticos 26, IM-UFRJ, Rio de Janeiro
1993
2
Cazenave, T.; Lions, P. L.
Orbital stability of standing waves for some nonlinear Schrödinger equations
Comm. Math. Phys.
85
1982
549-561
MR0677997
3
Fukuizumi, R.
Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential
Discrete Contin. Dynam. Systems
7
2001
525-544
MR1815766
4
Grillakis, M.; Shatah, J.; Strauss, W.
Stability theory of solitary waves in the presence of symmetry I
J. Funct. Anal.
74
1987
160-197
MR0901236
5
Hirose, M.; Ohta, M.
Structure of positive radial solutions to scalar field equations with harmonic potential
J. Diff. Eq.
178
2002
519-540
MR1879836
6
Kabeya, Y.; Tanaka, K.
Uniqueness of positive radial solutions of semilinear elliptic equations in $R^n$ and Séré's non-degeneracy condition
Comm. Partial Differential Equations
24
1999
563-598
MR1683050
7
Kusano, T.; Naito, M.
Oscillation theory of entire solutions of second order superlinear elliptic equations
Funkcial. Ekvac.
30
1987
269-282
MR0927184
0927184
8
Kwong, M. K.
Uniqueness of positive solutions of $\Delta u -u+u^p=0$ in $R^n$
Arch. Rational Mech. Anal.
105
1989
234-266
MR0969899
9
Li, Y.; Ni, W.-M.
Radial symmetry of positive solutions of nonlinear elliptic equations in $R^n$
Comm. Partial Differential Equations
18
1993
1043-1054
MR1218528
10
Rose, H. A.; Weinstein, M. I.
On the bound states of the nonlinear Schrödinger equation with a linear potential
Physica D
30
1988
207-218
MR0939275
11
Shatah, J.; Strauss, W.
Instability of nonlinear bound states
Comm. Math. Phys.
100
1985
173-190
MR0804458
12
Tsurumi, T.; Wadati, M.
Collapses of wavefunctions in multi-dimensional nonlinear Schrödinger equations under harmonic potential
J. Phys. Soc. Japan
66
1997
3031-3034
13
Weinstein, M. I.
Lyapunov stability of ground states of nonlinear dispersive evolution equations
Comm. Pure Appl. Math.
39
1986
51-68
MR0820338
14
Yanagida, E.
Uniqueness of positive radial solutions of $\Delta u+g(r)u+h(r)u^p=0$ in $R^n$
Arch. Rational Mech. Anal.
115
1991
257-274
MR1106294
15
Yanagida, E.; Yotsutani, S.
Classification of the structure of positive radial solutions to $\Delta u +K(|x|)u^{p}=0$ in $R^n$
Arch. Rational Mech. Anal.
124
1993
239-259
MR1237912
16
Yotsutani, S.
Pohozaev identity and its applications
Surikaisekikenkyusho Kokyuroku, Kyoto University, 834
1993
80-90
MR1283641
17
Zhang, J.
Stability of standing waves for nonlinear Schrödinger equations with unbounded potentials
Z. Angew. Math. Phys.
51
2000
489-503
MR1762704