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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.

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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