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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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

$\displaystyle \left \{ \begin {array}{l} u^{\prime \prime }(x)+\lambda f_{\vare... ...n }(u)=-\varepsilon u^{3}+\sigma u^{2}+\tau u+\rho \text {,}\end{array}\right .$    

where $ \lambda ,\varepsilon >0$ are two bifurcation parameters, and $ \sigma ,\rho >0,\tau \geq 0$ are constants. By developing some new time-map techniques, we prove the global bifurcation of bifurcation curves for varying $ \varepsilon >0$. More precisely, we prove that, for any $ \sigma ,\rho >0,\tau \geq 0$, there exists $ \varepsilon ^{\ast }>0$ such that, on the $ (\lambda ,\vert\vert u\vert\vert _{\infty })$-plane, the bifurcation curve is S-shaped for $ 0<\varepsilon <\varepsilon ^{\ast }$ and is monotone increasing for $ \varepsilon \geq \varepsilon ^{\ast }$. (We also prove the global bifurcation of bifurcation curves for varying $ \lambda >0$.) Thus we are able to determine the exact number of positive solutions by the values of $ \varepsilon $ and $ \lambda $. We give an application to prove a long-standing conjecture for global bifurcation of positive solutions for the problem

$\displaystyle \left \{ \begin {array}{l} u^{\prime \prime }(x)+\lambda (-\varep... ...+1)=0\text {, \ } -1<x<1\text {,} \\ u(-1)=u(1)=0\text {,} \end{array} \right .$    

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: http://dx.doi.org/10.1090/S0002-9947-2012-05670-4
PII: S 0002-9947(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.