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An elliptic boundary-value problem with a discontinuous nonlinearity, II

Published online by Cambridge University Press:  14 November 2011

G. Keady  and
P. E. Kloeden
G. Keady
Affiliation:
Mathematics Department, University of Western Australia, Nedlands, Western Australia 6009, Australia
P. E. Kloeden
Affiliation:
School of Mathematical and Physical Sciences, Murdoch University, Murdoch, Western Australia 6150, Australia
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Synopsis

Let Ω be a bounded domain in ℝ2. The study, begun in Keady [13], of the boundary-value problem, for (λ/k, ψ),

is continued. Here Δ denotes the Laplacian, H is the Heaviside step function and one of λ or k is a given positive constant. The solutions considered always have ψ > 0 in Ω and λ/k > 0, and have cores

In the special case Ω = B(0, R), a disc, the explicit exact solutions of the branch τe have connected cores A and the diameter of A tends to zero when the area of A tends to zero. This result is established here for other convex domains Ω and solutions with connected cores A.

An adaptation of the maximum principles and of the domain folding arguments of Gidas, Ni and Nirenberg [9] is an important step in establishing the above result.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 105 , Issue 1 , 1987 , pp. 23 - 36
Copyright
Copyright © Royal Society of Edinburgh 1987

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