Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-17T23:20:04.050Z Has data issue: false hasContentIssue false
  • English
  • Français

Symmetry breaking for Δu + 2δe−u = 0 on a disk with general boundary conditions

Published online by Cambridge University Press:  14 November 2011

Song-Sun Lin
Song-Sun Lin
Affiliation:
Department of Applied Mathematics, National Chiao Tung University, Hsin-Chu, Taiwan, Republic of China
Get access Rights & Permissions [Opens in a new window]

Synopsis

We discuss the radially, symmetric solutions and the symmetry breaking of the equation Δu + 2δe −u = 0 in D and u + b(∂u/∂n) = 0 on ∂D, where D is the unit disk in ℝ2, δ >0 and b is a constant. We prove that for any b < 0, there exists > 0 such that there are exactly two radially symmetric solutions for δ ∊ (0, ), one for δ = and none for δ > δ*b. For , where m is a positive integer, there are (b), k = 1, …, m, such that the equation has symmetry breaking at δ*k (b) on the lower branch of radially symmetric solutions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 113 , Issue 1-2 , 1989 , pp. 105 - 117
Copyright
Copyright © Royal Society of Edinburgh 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Cerami, G.. Symmetry breaking for a class of semilinear elliptic problems. Nonlinear Anal. 10 (1986), 1–14.CrossRefGoogle Scholar
2Cohen, D. S. and Laetsch, T. W.. Nonlinear boundary value problems suggested by chemical reactor theory. J. Differential Equations 7 (1970) 217–226.CrossRefGoogle Scholar
3Crandall, M. H. and Rabinowitz, P. H.. Bifurcation from simple eigenvalues. J. Fund. Anal. 8 (1971), 321–340.CrossRefGoogle Scholar
4Dancer, E. N.. On non-radially symmetric bifurcation. J. London Math. Soc. (2) 20 (1979), 287–292.Google Scholar
5Lin, S. S.. On non-radially symmetric bifurcation in the annulus. J. Differential Equations (to appear).Google Scholar
6 Liouville, J.. Sur l'équation aux dérivées partielles (∂2log λ)/(∂u ∂v) ±2λa 2 = 0. J. Math. 18 (1853), 71–72.Google Scholar
7Rabinowitz, P. H.. Some global results for non-linear eigenvalue problems. J. Fund. Anal. 7 (1971), 487–513.CrossRefGoogle Scholar
8Ramaswamy, M. and Srikanth, P. N.. Symmetry breaking for a class of semilinear elliptic problems. Trans. Amer. Math. Soc. 304 (1987), 839–845.CrossRefGoogle Scholar
9Smoller, J. and Wasserman, A.. Symmetry-breaking for positive solutions of semilinear elliptic equations. Arch. Rational Mech. Anal. 95 (1986), 217–225.CrossRefGoogle Scholar
10Smoller, J. and Wasserman, A.. Symmetry-breaking for solutions of semilinear elliptic equations with general boundary conditions. Commun. Math. Phys. 105 (1986), 415–441.CrossRefGoogle Scholar
11Suzuki, T. and Nagasaki, K.. On the non-linear eigenvalue problem Δu + λe u = 0. Trans. Amer. Math. Soc.(to appear).Google Scholar