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September  2018, 17(5): 2149-2171. doi: 10.3934/cpaa.2018103

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

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

Received  March 2017 Revised  January 2018 Published  April 2018

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.
Citation: Xuemei Zhang, Meiqiang Feng. Double bifurcation diagrams and four positive solutions of nonlinear boundary value problems via time maps. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2149-2171. doi: 10.3934/cpaa.2018103
References:
[1]

I. Addou and S.-H. Wang, Exact multiplicity results for a $p$-Laplacian problem with concave-convex-concave nonlinearities, Nonlinear Anal., 53 (2003), 111-137. 

[2]

N. Brubaker and J. Pelesko, Analysis of a one-dimensional prescribed mean curvature equation with singular nonlinearity, Nonlinear Anal., 75 (2012), 5086-5102. 

[3]

A. Castro and R. Shivaji, Non-negative solutions for a class of non-positone problems, Proc. Roy. Soc. Edinburgh Sec. A, 108 (1988), 291-302. 

[4]

J. Cheng, Exact number of positive solutions for a class of semipositone problems, J. Math. Anal. Appl., 280 (2003), 197-208. 

[5]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. 

[6]

J. Goddard IIE. K. Lee and R. Shivaji, A double $S$-shaped bifurcation curve for a reaction-diffusion model with nonlinear boundary conditions, Bound. Value. Probl., 2010 (2010), 357542. 

[7]

K.-C. HungY.-H. ChengS.-H. Wang and C.-H. Chuang, Exact multiplicity and bifurcation diagrams of positive solutions of a one-dimensional multiparameter prescribed mean curvature problem, J. Differential Equations, 257 (2014), 3272-3299. 

[8]

K.-C. Hung and S.-H. Wang, A theorem on S-shaped bifurcation curve for a positone problem with convex-concave nonlinearity and its applications to the perturbed Gelfand problem, J. Differential Equations, 251 (2011), 223-237. 

[9]

S.-Y. Huang and S.-H. Wang, Proof of a conjecture for the one-dimensional perturbed Gelfand problem from combustion theory, Arch. Ration. Mech. An., 222 (2016), 769-825. 

[10]

K.-C. HungS.-H. Wang and C.-H. Yu, Existence of a double $S$-shaped bifurcation curve with six positive solutions for a combustion problem, J. Math. Anal. Appl., 392 (2012), 40-54. 

[11]

D. Joseph and T. Lundgren, Quasilinear Dirichlet problems driven by positive source, Arch. Rational Mech. Anal., 49 (1973), 241-269. 

[12]

P. Korman, Uniqueness and exact multiplicity of solutions for a class of Dirichlet problems, J. Differential Equations, 244 (2008), 2602-2613. 

[13]

P. Korman and Y. Li, On the exactness of an $S$-shaped bifurcation curve, Proc. Amer. Math. Soc., 127 (1999), 1011-1020. 

[14]

T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1-13. 

[15]

Y.-H. Liang and S.-H. Wang, Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions, J. Differential Equations, 260 (2016), 8358-8387. 

[16]

Y.-H. Liang and S.-H. Wang, Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions Ⅱ, Electron. J. Differential Equations, 61 (2017), 1-12. 

[17]

J. Liouville, Sur léquation aux différences partielles $\frac{d^2\logλ}{dudv}± \frac{λ}{2a^2}=0$, J. Math. Pures Appl., 18 (1853), 71-72. 

[18]

Z. Liu and X. Zhang, A class of two-point boundary value problems, J. Math. Anal. Appl., 254 (2001), 599-617. 

[19]

H. Pan and R. Xing, Exact multiplicity results for a one-dimensional prescribed mean curvature problem related to MEMS models, Nonlinear Anal. Real World Appl., 13 (2012), 2432-2445. 

[20]

H. Pan and R. Xing, On the existence of positive solutions for some nonlinear boundary value problems and applications to MEMS model, Discret. Contin. Dyn. Syst., 35 (2015), 3627-3682. 

[21]

J. Shi and R. Shivaji, Exact multiplicity of solutions for classes of semipositone problems with concave-convex nonlinearity, Discret Contin. Dyn. Syst., 7 (2001), 559-571. 

[22]

J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations, 39 (1981), 269-290. 

[23]

S.-H. Wang and T.-S. Yeh, Exact multiplicity and ordering properties of positive solutions of a $p$-Laplacian Dirichlet problem and their applications, J. Math. Anal. Appl., 287 (2003), 380-398. 

[24]

X. Zhang and M. Feng, Exact number of solutions of a one-dimensional prescribed mean curvature equation with concave-convex nonlinearities, J. Math. Anal. Appl., 395 (2012), 393-402. 

show all references

References:
[1]

I. Addou and S.-H. Wang, Exact multiplicity results for a $p$-Laplacian problem with concave-convex-concave nonlinearities, Nonlinear Anal., 53 (2003), 111-137. 

[2]

N. Brubaker and J. Pelesko, Analysis of a one-dimensional prescribed mean curvature equation with singular nonlinearity, Nonlinear Anal., 75 (2012), 5086-5102. 

[3]

A. Castro and R. Shivaji, Non-negative solutions for a class of non-positone problems, Proc. Roy. Soc. Edinburgh Sec. A, 108 (1988), 291-302. 

[4]

J. Cheng, Exact number of positive solutions for a class of semipositone problems, J. Math. Anal. Appl., 280 (2003), 197-208. 

[5]

M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. 

[6]

J. Goddard IIE. K. Lee and R. Shivaji, A double $S$-shaped bifurcation curve for a reaction-diffusion model with nonlinear boundary conditions, Bound. Value. Probl., 2010 (2010), 357542. 

[7]

K.-C. HungY.-H. ChengS.-H. Wang and C.-H. Chuang, Exact multiplicity and bifurcation diagrams of positive solutions of a one-dimensional multiparameter prescribed mean curvature problem, J. Differential Equations, 257 (2014), 3272-3299. 

[8]

K.-C. Hung and S.-H. Wang, A theorem on S-shaped bifurcation curve for a positone problem with convex-concave nonlinearity and its applications to the perturbed Gelfand problem, J. Differential Equations, 251 (2011), 223-237. 

[9]

S.-Y. Huang and S.-H. Wang, Proof of a conjecture for the one-dimensional perturbed Gelfand problem from combustion theory, Arch. Ration. Mech. An., 222 (2016), 769-825. 

[10]

K.-C. HungS.-H. Wang and C.-H. Yu, Existence of a double $S$-shaped bifurcation curve with six positive solutions for a combustion problem, J. Math. Anal. Appl., 392 (2012), 40-54. 

[11]

D. Joseph and T. Lundgren, Quasilinear Dirichlet problems driven by positive source, Arch. Rational Mech. Anal., 49 (1973), 241-269. 

[12]

P. Korman, Uniqueness and exact multiplicity of solutions for a class of Dirichlet problems, J. Differential Equations, 244 (2008), 2602-2613. 

[13]

P. Korman and Y. Li, On the exactness of an $S$-shaped bifurcation curve, Proc. Amer. Math. Soc., 127 (1999), 1011-1020. 

[14]

T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1-13. 

[15]

Y.-H. Liang and S.-H. Wang, Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions, J. Differential Equations, 260 (2016), 8358-8387. 

[16]

Y.-H. Liang and S.-H. Wang, Classification and evolution of bifurcation curves for the one-dimensional perturbed Gelfand equation with mixed boundary conditions Ⅱ, Electron. J. Differential Equations, 61 (2017), 1-12. 

[17]

J. Liouville, Sur léquation aux différences partielles $\frac{d^2\logλ}{dudv}± \frac{λ}{2a^2}=0$, J. Math. Pures Appl., 18 (1853), 71-72. 

[18]

Z. Liu and X. Zhang, A class of two-point boundary value problems, J. Math. Anal. Appl., 254 (2001), 599-617. 

[19]

H. Pan and R. Xing, Exact multiplicity results for a one-dimensional prescribed mean curvature problem related to MEMS models, Nonlinear Anal. Real World Appl., 13 (2012), 2432-2445. 

[20]

H. Pan and R. Xing, On the existence of positive solutions for some nonlinear boundary value problems and applications to MEMS model, Discret. Contin. Dyn. Syst., 35 (2015), 3627-3682. 

[21]

J. Shi and R. Shivaji, Exact multiplicity of solutions for classes of semipositone problems with concave-convex nonlinearity, Discret Contin. Dyn. Syst., 7 (2001), 559-571. 

[22]

J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations, 39 (1981), 269-290. 

[23]

S.-H. Wang and T.-S. Yeh, Exact multiplicity and ordering properties of positive solutions of a $p$-Laplacian Dirichlet problem and their applications, J. Math. Anal. Appl., 287 (2003), 380-398. 

[24]

X. Zhang and M. Feng, Exact number of solutions of a one-dimensional prescribed mean curvature equation with concave-convex nonlinearities, J. Math. Anal. Appl., 395 (2012), 393-402. 

Figure 1.  Reversed $S$-shaped curve
Figure 2.  Broken reversed $S$-shaped curve
Figure 3.  Exactly $S$-shaped bifurcation curve $S$ of $(1.6)$
Figure 6.  Bifurcation diagram of $C = S\bigcup\widetilde{S}$ of (1.1) in the case $f(u) = e^{u}$
Figure 8.  Bifurcation diagrams of $C = S\bigcup\widetilde{S}$ of (1.1) in the case $f(u) = a^{u}$
Figure 10.  Bifurcation diagrams of $C = S\bigcup\widetilde{S}$ of (1.1) in the case $f(u) = u^p$
Figure 12.  Bifurcation diagram of $C = S\bigcup\widetilde{S}$ of (1.1) in the case $f(u) = e^{u}-1$
Figure 14.  Bifurcation diagram of $C = S\bigcup\widetilde{S}$ of (1.1) in the case $f(u) = (1+u)^p(p>1)$
Figure 16.  Bifurcation diagrams of $C = S\bigcup\widetilde{S}$ of (1.1) in the case $f(u) = (1+u)^p(p\leq 1)$
Figure 4.  Graph of $G(\lambda,\rho)$ for fixed $\lambda$ in the case $f(u) = e^{u}$
Figure 5.  Graphs of $G(\lambda,\rho)$ and $H(\lambda,\rho)$ for $\lambda_{0},\lambda_{1},\lambda_{2}$ in the case $f(u) = e^{u}$
Figure 7.  Graphs of $G(\lambda,\rho)$ for fixed $\lambda$ in the case $f(u) = a^{u}$
Figure 9.  Graphs of $G(\lambda,\rho)$ for fixed $\lambda$ in the case $f(u) = u^p$
Figure 11.  Graph of $G(\lambda,\rho)$ for fixed $\lambda$ in the case $f(u) = e^{u}-1$
Figure 13.  Graphs of $G(\lambda,\rho)$ for fixed $\lambda$ in the case $f(u) = (1+u)^p(p>0)$
Figure 15.  Graphs of $\widetilde{H}(\rho,0).$
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