Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications

Authors:
Kuo-Chih Hung and Shin-Hwa Wang

Journal:
Trans. Amer. Math. Soc. **365** (2013), 1933-1956

MSC (2010):
Primary 34B18, 74G35

Published electronically:
August 22, 2012

MathSciNet review:
3009649

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Abstract | References | Similar Articles | Additional Information

Abstract: We study the global bifurcation and exact multiplicity of positive solutions of

where are two bifurcation parameters, and are constants. By developing some new time-map techniques, we prove the global bifurcation of bifurcation curves for varying . More precisely, we prove that, for any , there exists such that, on the -plane, the bifurcation curve is S-shaped for and is monotone increasing for . (We also prove the global bifurcation of bifurcation curves for varying .) Thus we are able to determine the exact number of positive solutions by the values of and . We give an application to prove a long-standing conjecture for global bifurcation of positive solutions for the problem

which was studied by Crandall and Rabinowitz (Arch. Rational Mech. Anal. 52 (1973), p. 177). In addition, we give an application to prove a conjecture of Smoller and Wasserman (J. Differential Equations 39 (1981), p. 283, lines 2-3) on the maximum number of positive solutions of a positone problem.

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Additional Information

**Kuo-Chih Hung**

Affiliation:
Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 300, Republic of China

Email:
kchung@mx.nthu.edu.tw

**Shin-Hwa Wang**

Affiliation:
Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 300, Republic of China

Email:
shwang@math.nthu.edu.tw

DOI:
https://doi.org/10.1090/S0002-9947-2012-05670-4

Keywords:
Global bifurcation,
exact multiplicity,
positive solution,
positone problem,
S-shaped bifurcation curve,
time map.

Received by editor(s):
July 9, 2010

Received by editor(s) in revised form:
February 19, 2011, and June 23, 2011

Published electronically:
August 22, 2012

Additional Notes:
This work was partially supported by the National Science Council of the Republic of China under grant No. 98-2115-M-007-008-MY3.

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.