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The set of positive solutions of semilinear equations in large balls

Published online by Cambridge University Press:  14 November 2011

R. Gardner  and
L. A. Peletier
R. Gardner
Affiliation:
Department of Mathematics, University of Massachusetts, Amherst, MA 01003, U.S.A.
L. A. Peletier
Affiliation:
Mathematical Institute, University of Leiden, Niels Bohrweg 1, Postbus 9512, 2300 RA Leiden, The Netherlands
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Synopsis

The exact number of positive solutions of Δu + f(u) = 0 on finite balls in ℝN is determined. The assumptions about f(u) are similar to those imposed by Serrin and the second author in a previous study of uniqueness of the positive solution when the spatial domain is all of ℝN (see [7, 8]). For finite balls of sufficiently large radius it is shown here that there are exactly two positive and, hence, radial solutions. To this end, we first prove the linear nondegeneracy of the positive solution of ℝN. This is obtained by applying the technique of monotone separation of graphs [7] to the linearised equations. Somewhat sharper estimates are required here (see Part I, Section 2).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 104 , Issue 1-2 , 1986 , pp. 53 - 72
Copyright
Copyright © Royal Society of Edinburgh 1986

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