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A semilinear elliptic eigenvalue problem, I

Published online by Cambridge University Press:  14 November 2011

Grant Keady  and
John Norbury
Grant Keady
Affiliation:
Mathematics Department, University of Western Australia
John Norbury
Affiliation:
St Catherine's College, Oxford
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Synopsis

The nonlinear operator equation (N), , describes both semilinear elliptic boundary value problems (P) and their natural discretisations (Ph). (Here is a positive compact linear operator and [t]+ ≡ (t + |t|)/2 for all t ∊ ℝ.) It is proved that, for q ≧ 0 (q ≢ 0), in a Banach lattice E the equation (N) has an unbounded continuum ℱ of nontrivial solutions (λ, ψ) ∊ ℝ × E bifurcating from infinity at (λ1, ∞) (here is the first (positive) eigenvalue of ). All nontrivial solutions (λ, ψ) have λ ≧ λ1, and if maps a smaller cone Ks, into itself then there is a ℱs ⊂ℝ×Ks with similar properties to ℱ. The existence theory for (N) is applied to problems (P) and (Ph) which are defined on the simply-connected domain Ω. It is shown that the projection of ℱ on the λ-axis is either unbounded if the continuous function q > 0 except on a set of measure zero in Ω, or is bounded if q ≡ 0 on a subdomain of Ω. If is the second eigenvalue for in (Ph) then there is at most one nontrivial solution for each λ satisfying λ1 < λ < λ2; in the corresponding uniqueness result for (P) q is restricted to being strictly positive somewhere on ∂Ω Additional properties for the solutions of (P) and (Ph) are also proved.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 87 , Issue 1-2 , 1980 , pp. 65 - 82
Copyright
Copyright © Royal Society of Edinburgh 1980

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