Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-18T21:52:53.415Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Adams, R.. Sobolev Spaces (New York: Academic Press, 1975).Google Scholar
2Ambrosetti, A., Arcoya, D. and Buffoni, B.. Positive solutions for some semipositone problems via Bifurcation theory. Differential Integral Equations 7 (1994), 655–64.CrossRefGoogle Scholar
3Castro, A. and Gadam, S.. Uniqueness of stable and unstable positive solutions for semipositone problems. Nonlinear Anal. 22 (1994), 425–9.CrossRefGoogle Scholar
4Castro, A., Gadam, S. and Shivaji, R.. Branches of radial solutions for semipositone problems. J. Differential Equations 120 (1995), 30–45.CrossRefGoogle Scholar
5Castro, A., Gadam, S. and Shivaji, R.. Evolution of positive solution curves in semipositone problems with concave nonlinearities (Preprint).Google Scholar
6Castro, A., Garner, J. and Shivaji, R.. Existence results for classes of sublinear semipositone problems. Results Math. 23 (1993), 214–20.CrossRefGoogle Scholar
7Castro, A. and Shivaji, R.. Positive solutions for a concave semipositone Dirichlet problem. Nonlinear Anal., (to appear).Google Scholar
8Crandall, M. G. and Rabinowitz, P. H.. Bifurcation from simple eigenvalues. J. Funct. Anal. 8 (1971), 321–40.CrossRefGoogle Scholar
9Wang, S.-H.. Positive solutions for a class of nonpositone problems with concave nonlinearities. Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), 507–15.CrossRefGoogle Scholar