Double bifurcation diagrams and four positive solutions of nonlinear boundary value problems via time maps

Xuemei Zhang; Meiqiang Feng

doi:10.3934/cpaa.2018103

Article Contents

2018, Volume 17, Issue 5: 2149-2171. Doi: 10.3934/cpaa.2018103

This issue Previous Article Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms Next Article Specified homogenization of a discrete traffic model leading to an effective junction condition

Double bifurcation diagrams and four positive solutions of nonlinear boundary value problems via time maps

Xuemei Zhang^1, , and
Meiqiang Feng^2,

1.
Department of Mathematics and Physics, North China Electric Power University, Beijing, 102206, China
2.
School of Applied Science, Beijing Information Science & Technology University, Beijing, 100192, China

* Corresponding author
* Corresponding author

Received: February 28, 2017

Revised: December 31, 2017

Published: April 2018

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Abstract

In this paper, we consider the existence and exactness of multiple positive solutions for the nonlinear boundary value problem

$\left\{ \begin{array}{l} - u''(x) = \lambda f(u),\;\;\;\;0 < x < 1,\\u(0) = 0,\\\frac{{u(1)}}{{u(1) + 1}}u'(1) + \left[ {1 - \frac{{u(1)}}{{u(1) + 1}}} \right]u(1) = 0,\end{array} \right.$

where $λ>0$ is a bifurcation parameter, $f(u)>0$ for $u>0$. We give complete descriptions of the structure of bifurcation curves and determine the existence and multiplicity of positive solutions of the above problem for $f(u) = e^{u},\ f(u) = a^{u}(a>0),\ f(u) = u^{p}(p>0),\ f(u) = e^{u}-1,\ f(u) = a^{u}-1(a>1)$ and $f(u) = (1+u)^{p}(p>0)$. Our methods are based on a detailed analysis of time maps.

Keywords:

Double bifurcation diagrams,
existence and multiplicity of positive solutions,
nonlinear boundary conditions,
time map.

Mathematics Subject Classification: 34B18, 74G35.

Citation:

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Figure 1. Reversed $S$-shaped curve

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Figure 2. Broken reversed $S$-shaped curve

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Figure 3. Exactly $S$-shaped bifurcation curve $S$ of $(1.6)$

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Figure 6. Bifurcation diagram of $C = S\bigcup\widetilde{S}$ of (1.1) in the case $f(u) = e^{u}$

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Figure 8. Bifurcation diagrams of $C = S\bigcup\widetilde{S}$ of (1.1) in the case $f(u) = a^{u}$

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Figure 10. Bifurcation diagrams of $C = S\bigcup\widetilde{S}$ of (1.1) in the case $f(u) = u^p$

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Figure 12. Bifurcation diagram of $C = S\bigcup\widetilde{S}$ of (1.1) in the case $f(u) = e^{u}-1$

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Figure 14. Bifurcation diagram of $C = S\bigcup\widetilde{S}$ of (1.1) in the case $f(u) = (1+u)^p(p>1)$

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Figure 16. Bifurcation diagrams of $C = S\bigcup\widetilde{S}$ of (1.1) in the case $f(u) = (1+u)^p(p\leq 1)$

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Figure 4. Graph of $G(\lambda,\rho)$ for fixed $\lambda$ in the case $f(u) = e^{u}$

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Figure 5. Graphs of $G(\lambda,\rho)$ and $H(\lambda,\rho)$ for $\lambda_{0},\lambda_{1},\lambda_{2}$ in the case $f(u) = e^{u}$

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Figure 7. Graphs of $G(\lambda,\rho)$ for fixed $\lambda$ in the case $f(u) = a^{u}$

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Figure 9. Graphs of $G(\lambda,\rho)$ for fixed $\lambda$ in the case $f(u) = u^p$

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Figure 11. Graph of $G(\lambda,\rho)$ for fixed $\lambda$ in the case $f(u) = e^{u}-1$

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Figure 13. Graphs of $G(\lambda,\rho)$ for fixed $\lambda$ in the case $f(u) = (1+u)^p(p>0)$

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Figure 15. Graphs of $\widetilde{H}(\rho,0).$

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