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Positive solution curves of semipositone problems with concave nonlinearities*

Published online by Cambridge University Press:  14 November 2011

Alfonso Castro ,
Sudhasree Gadam  and
R. Shivaji
Alfonso Castro
Affiliation:
Department of Mathematics, University of North Texas, Denton, TX 76203-5116, U.S.A.
Sudhasree Gadam
Affiliation:
Mathematics, Yashodha, J.C.R. VI Cross, Chitradurga, India 577501
R. Shivaji
Affiliation:
Department of Mathematics, Mississippi State University, Mississippi State, MS 39762, U.S.A.
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Synopsis

We consider the positive solutions to the semilinear equation:

where Ω denotes a smooth bounded region in ℝN(N > 1) and ℷ 0. Here f :[0, ∞)→ℝ is assumed to be monotonically increasing, concave and such that f(0)<0 (semipositone). Assuming that f′(∞)≡limt→∞f(t)> 0, we establish the stability and uniqueness of large positive solutions in terms of (f(t)/t)′ When Ω is a ball, we determine the exact number of positive solutions for each λ > 0. We also obtain the geometry of the branches of positive solutions completely and establish how they evolve. This work extends and complements that of [3, 7] where f′(∞)≦0.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 5 , 1997 , pp. 921 - 934
Copyright
Copyright © Royal Society of Edinburgh 1997

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