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Padé approximant bounds on the positive solutions of some nonlinear elliptic equations

Published online by Cambridge University Press:  14 November 2011

M. F. Barnsley  and
D. Bessis
M. F. Barnsley
Affiliation:
School of Mathematics, University of Bradford
D. Bessis
Affiliation:
Service de Physique Théorique, C.E.N.S., B.P. No. 2, 91190 Gif-sur-Yvette, France
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Synopsis

We consider the equation Lφ − λpφ + γqφ2 = f on a bounded domain in Rn with homogeneous Neumann-Dirichlet boundary conditions. L is a negative definite uniformly elliptic differential operator, while, p, q and f are positive functions. We show that there exists exactly one positive solution for each λ ∈ R and γ > 0. This solution can be analytically continued throughout Re γ > 0: it is a Laplace transform of a positive measure. The measure is bounded prior to the bifurcation point of the associated “homogeneous” equation and unbounded after. Noting that any Laplace transform of positive measure has associated with it a natural sequence of Tchebycheff systems, it now follows that one can obtain monotonically converging upper and lower bounds which are provided by the generalized Padé approximants generated from the Tchebycheff systems.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 80 , Issue 3-4 , 1978 , pp. 183 - 200
Copyright
Copyright © Royal Society of Edinburgh 1978

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