Global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications
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- by Kuo-Chih Hung and Shin-Hwa Wang PDF
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Abstract:
We study the global bifurcation and exact multiplicity of positive solutions of \begin{equation*} \left \{ \begin {array}{l} u^{\prime \prime }(x)+\lambda f_{\varepsilon }(u)=0\text {,}\;\;-1<x<1\text {, \ }u(-1)=u(1)=0\text {,} \\ f_{\varepsilon }(u)=-\varepsilon u^{3}+\sigma u^{2}+\tau u+\rho \text {,}\end{array}\right . \end{equation*} 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 $(\lambda ,||u||_{\infty })$-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 \begin{equation*} \left \{ \begin {array}{l} u^{\prime \prime }(x)+\lambda (-\varepsilon u^{3}+u^{2}+u+1)=0\text {, \ } -1<x<1\text {,} \\ u(-1)=u(1)=0\text {,} \end{array} \right . \end{equation*} 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.References
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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
- 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.
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 1933-1956
- MSC (2010): Primary 34B18, 74G35
- DOI: https://doi.org/10.1090/S0002-9947-2012-05670-4
- MathSciNet review: 3009649