Axial symmetry of some steady state solutions to nonlinear Schrödinger equations

Authors:
Changfeng Gui, Andrea Malchiodi and Haoyuan Xu

Journal:
Proc. Amer. Math. Soc. **139** (2011), 1023-1032

MSC (2010):
Primary 35J15, 35J20, 35J60, 35J61

Published electronically:
September 1, 2010

MathSciNet review:
2745653

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we show the axial symmetry of steady state solutions of nonlinear Schrodinger equations when the exponent of the nonlinearity is between the critical Sobolev exponent of -dimensional space and -dimensional space.

**1.**Jérôme Busca and Patricio Felmer,*Qualitative properties of some bounded positive solutions to scalar field equations*, Calc. Var. Partial Differential Equations**13**(2001), no. 2, 191–211. MR**1861097**, 10.1007/PL00009928**2.**E. N. Dancer,*On the uniqueness of the positive solution of a singularly perturbed problem*, Rocky Mountain J. Math.**25**(1995), no. 3, 957–975. MR**1357103**, 10.1216/rmjm/1181072198**3.**Edward Norman Dancer,*New solutions of equations on 𝐑ⁿ*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)**30**(2001), no. 3-4, 535–563 (2002). MR**1896077****4.**M. Del Pino, M. Kowalzyck, F. Pacard and J. Wei, The Toda system and multiple-end solutions of autonomous planar elliptic problems, preprint, 2007.**5.**Changfeng Gui,*Hamiltonian identities for elliptic partial differential equations*, J. Funct. Anal.**254**(2008), no. 4, 904–933. MR**2381198**, 10.1016/j.jfa.2007.10.015**6.**Changfeng Gui,*Hamiltonian constants for several new entire solutions*, Front. Math. China**3**(2008), no. 2, 195–204. MR**2395216**, 10.1007/s11464-008-0012-2**7.**C. Gui, Symmetry of traveling wave solutions to the Allen-Cahn equation in , preprint.**8.**C. Gui, Symmetry of certain saddle solutions to the Allen-Cahn equation, preprint.**9.**Andrea Malchiodi, Some new entire solutions of semilinear elliptic equations on , preprint, 2007.

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Additional Information

**Changfeng Gui**

Affiliation:
Department of Mathematics, U-9, University of Connecticut Storrs, Connecticut 06269

Email:
gui@math.uconn.edu

**Andrea Malchiodi**

Affiliation:
Sector of Mathematical Analysis, SISSA, Via Beirut 2-4, 34014 Trieste, Italy

Email:
malchiod@sissa.it

**Haoyuan Xu**

Affiliation:
Department of Mathematics, U-9, University of Connecticut, Storrs, Connecticut 06269

Email:
haoyuan@math.uconn.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-2010-10638-X

Keywords:
Nonlinear Schrödinger equation,
steady state solution,
Hamiltonian identity,
axial symmetry,
the moving plane method

Received by editor(s):
March 28, 2010

Published electronically:
September 1, 2010

Communicated by:
Matthew J. Gursky

Article copyright:
© Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.