Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications
Authors:
KuoChih Hung and ShinHwa Wang
Journal:
Trans. Amer. Math. Soc. 365 (2013), 19331956
MSC (2010):
Primary 34B18, 74G35
Published electronically:
August 22, 2012
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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 timemap 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 Sshaped 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 longstanding 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 23) on the maximum number of positive solutions of a positone problem.
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Additional Information
KuoChih Hung
Affiliation:
Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 300, Republic of China
Email:
kchung@mx.nthu.edu.tw
ShinHwa Wang
Affiliation:
Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 300, Republic of China
Email:
shwang@math.nthu.edu.tw
DOI:
http://dx.doi.org/10.1090/S000299472012056704
PII:
S 00029947(2012)056704
Keywords:
Global bifurcation,
exact multiplicity,
positive solution,
positone problem,
Sshaped 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. 982115M007008MY3.
Article copyright:
© Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
