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Non-negative solutions for a class of non-positone problems

Published online by Cambridge University Press:  14 November 2011

Alfonso Castro  and
R. Shivaji
Alfonso Castro
Affiliation:
Department of Mathematics, North Texas State University, Denton, TX 76203–5116, U.S.A.
R. Shivaji
Affiliation:
Department of Mathematics, Mississippi State University, Mississippi State, MS 39762, U.S.A.
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Synopsis

In the recent past many results have been established on non-negative solutions to boundary value problems of the form

where λ>0, f(0)>0 (positone problems). In this paper we consider the impact on the non-negative solutions when f(0)<0. We find that we need f(u) to be convex to guarantee uniqueness of positive solutions, and f(u) to be appropriately concave for multiple positive solutions. This is in contrast to the case of positone problems, where the roles of convexity and concavity were interchanged to obtain similar results. We further establish the existence of non-negative solutions with interior zeros, which did not exist in positone problems.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 108 , Issue 3-4 , 1988 , pp. 291 - 302
Copyright
Copyright © Royal Society of Edinburgh 1988

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References

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