Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 

 

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
Full-text PDF

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 $ n$-dimensional space and $ (n-1)$-dimensional space.


References [Enhancements On Off] (What's this?)

  • 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 $ \mathbb{R}^2$, 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 $ \mathbb{R}^n$, preprint, 2007.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35J15, 35J20, 35J60, 35J61

Retrieve articles in all journals with MSC (2010): 35J15, 35J20, 35J60, 35J61


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.