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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
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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.

Information

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

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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