# American Institute of Mathematical Sciences

• Previous Article
Specified homogenization of a discrete traffic model leading to an effective junction condition
• CPAA Home
• This Issue
• Next Article
Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms
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 & 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.   Google Scholar [2] N. Brubaker and J. Pelesko, Analysis of a one-dimensional prescribed mean curvature equation with singular nonlinearity, Nonlinear Anal., 75 (2012), 5086-5102.   Google Scholar [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.   Google Scholar [4] J. Cheng, Exact number of positive solutions for a class of semipositone problems, J. Math. Anal. Appl., 280 (2003), 197-208.   Google Scholar [5] M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.   Google Scholar [6] J. Goddard II, E. 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.   Google Scholar [7] K.-C. Hung, Y.-H. Cheng, S.-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.   Google Scholar [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.   Google Scholar [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.   Google Scholar [10] K.-C. Hung, S.-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.   Google Scholar [11] D. Joseph and T. Lundgren, Quasilinear Dirichlet problems driven by positive source, Arch. Rational Mech. Anal., 49 (1973), 241-269.   Google Scholar [12] P. Korman, Uniqueness and exact multiplicity of solutions for a class of Dirichlet problems, J. Differential Equations, 244 (2008), 2602-2613.   Google Scholar [13] P. Korman and Y. Li, On the exactness of an $S$-shaped bifurcation curve, Proc. Amer. Math. Soc., 127 (1999), 1011-1020.   Google Scholar [14] T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1-13.   Google Scholar [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.   Google Scholar [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.   Google Scholar [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.   Google Scholar [18] Z. Liu and X. Zhang, A class of two-point boundary value problems, J. Math. Anal. Appl., 254 (2001), 599-617.   Google Scholar [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.   Google Scholar [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.   Google Scholar [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.   Google Scholar [22] J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations, 39 (1981), 269-290.   Google Scholar [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.   Google Scholar [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.   Google Scholar

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.   Google Scholar [2] N. Brubaker and J. Pelesko, Analysis of a one-dimensional prescribed mean curvature equation with singular nonlinearity, Nonlinear Anal., 75 (2012), 5086-5102.   Google Scholar [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.   Google Scholar [4] J. Cheng, Exact number of positive solutions for a class of semipositone problems, J. Math. Anal. Appl., 280 (2003), 197-208.   Google Scholar [5] M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180.   Google Scholar [6] J. Goddard II, E. 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.   Google Scholar [7] K.-C. Hung, Y.-H. Cheng, S.-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.   Google Scholar [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.   Google Scholar [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.   Google Scholar [10] K.-C. Hung, S.-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.   Google Scholar [11] D. Joseph and T. Lundgren, Quasilinear Dirichlet problems driven by positive source, Arch. Rational Mech. Anal., 49 (1973), 241-269.   Google Scholar [12] P. Korman, Uniqueness and exact multiplicity of solutions for a class of Dirichlet problems, J. Differential Equations, 244 (2008), 2602-2613.   Google Scholar [13] P. Korman and Y. Li, On the exactness of an $S$-shaped bifurcation curve, Proc. Amer. Math. Soc., 127 (1999), 1011-1020.   Google Scholar [14] T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1-13.   Google Scholar [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.   Google Scholar [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.   Google Scholar [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.   Google Scholar [18] Z. Liu and X. Zhang, A class of two-point boundary value problems, J. Math. Anal. Appl., 254 (2001), 599-617.   Google Scholar [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.   Google Scholar [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.   Google Scholar [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.   Google Scholar [22] J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations, 39 (1981), 269-290.   Google Scholar [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.   Google Scholar [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.   Google Scholar
Reversed $S$-shaped curve
Broken reversed $S$-shaped curve
Exactly $S$-shaped bifurcation curve $S$ of $(1.6)$
Bifurcation diagram of $C = S\bigcup\widetilde{S}$ of (1.1) in the case $f(u) = e^{u}$
Bifurcation diagrams of $C = S\bigcup\widetilde{S}$ of (1.1) in the case $f(u) = a^{u}$
Bifurcation diagrams of $C = S\bigcup\widetilde{S}$ of (1.1) in the case $f(u) = u^p$
Bifurcation diagram of $C = S\bigcup\widetilde{S}$ of (1.1) in the case $f(u) = e^{u}-1$
Bifurcation diagram of $C = S\bigcup\widetilde{S}$ of (1.1) in the case $f(u) = (1+u)^p(p>1)$
Bifurcation diagrams of $C = S\bigcup\widetilde{S}$ of (1.1) in the case $f(u) = (1+u)^p(p\leq 1)$
Graph of $G(\lambda,\rho)$ for fixed $\lambda$ in the case $f(u) = e^{u}$
Graphs of $G(\lambda,\rho)$ and $H(\lambda,\rho)$ for $\lambda_{0},\lambda_{1},\lambda_{2}$ in the case $f(u) = e^{u}$
Graphs of $G(\lambda,\rho)$ for fixed $\lambda$ in the case $f(u) = a^{u}$
Graphs of $G(\lambda,\rho)$ for fixed $\lambda$ in the case $f(u) = u^p$
Graph of $G(\lambda,\rho)$ for fixed $\lambda$ in the case $f(u) = e^{u}-1$
Graphs of $G(\lambda,\rho)$ for fixed $\lambda$ in the case $f(u) = (1+u)^p(p>0)$
Graphs of $\widetilde{H}(\rho,0).$
 [1] Michael E. Filippakis, Nikolaos S. Papageorgiou. Existence and multiplicity of positive solutions for nonlinear boundary value problems driven by the scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2004, 3 (4) : 729-756. doi: 10.3934/cpaa.2004.3.729 [2] Santiago Cano-Casanova. Bifurcation to positive solutions in BVPs of logistic type with nonlinear indefinite mixed boundary conditions. Conference Publications, 2013, 2013 (special) : 95-104. doi: 10.3934/proc.2013.2013.95 [3] Tai-Chia Lin, Tsung-Fang Wu. Existence and multiplicity of positive solutions for two coupled nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2911-2938. doi: 10.3934/dcds.2013.33.2911 [4] Gennaro Infante. Positive solutions of differential equations with nonlinear boundary conditions. Conference Publications, 2003, 2003 (Special) : 432-438. doi: 10.3934/proc.2003.2003.432 [5] Trad Alotaibi, D. D. Hai, R. Shivaji. Existence and nonexistence of positive radial solutions for a class of $p$-Laplacian superlinear problems with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4655-4666. doi: 10.3934/cpaa.2020131 [6] Mohan Mallick, Sarath Sasi, R. Shivaji, S. Sundar. Bifurcation, uniqueness and multiplicity results for classes of reaction diffusion equations arising in ecology with nonlinear boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021195 [7] Julián López-Góme, Andrea Tellini, F. Zanolin. High multiplicity and complexity of the bifurcation diagrams of large solutions for a class of superlinear indefinite problems. Communications on Pure & Applied Analysis, 2014, 13 (1) : 1-73. doi: 10.3934/cpaa.2014.13.1 [8] Dagny Butler, Eunkyung Ko, Eun Kyoung Lee, R. Shivaji. Positive radial solutions for elliptic equations on exterior domains with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2713-2731. doi: 10.3934/cpaa.2014.13.2713 [9] Tsung-Fang Wu. Multiplicity of positive solutions for a semilinear elliptic equation in $R_+^N$ with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1675-1696. doi: 10.3934/cpaa.2010.9.1675 [10] Eric R. Kaufmann. Existence and nonexistence of positive solutions for a nonlinear fractional boundary value problem. Conference Publications, 2009, 2009 (Special) : 416-423. doi: 10.3934/proc.2009.2009.416 [11] Daniel Franco, Donal O'Regan. Existence of solutions to second order problems with nonlinear boundary conditions. Conference Publications, 2003, 2003 (Special) : 273-280. doi: 10.3934/proc.2003.2003.273 [12] Shao-Yuan Huang. Bifurcation diagrams of positive solutions for one-dimensional Minkowski-curvature problem and its applications. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 3443-3462. doi: 10.3934/dcds.2019142 [13] Julián López-Gómez, Marcela Molina-Meyer, Paul H. Rabinowitz. Global bifurcation diagrams of one node solutions in a class of degenerate boundary value problems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 923-946. doi: 10.3934/dcdsb.2017047 [14] Jiafeng Liao, Peng Zhang, Jiu Liu, Chunlei Tang. Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1959-1974. doi: 10.3934/dcdss.2016080 [15] Leonelo Iturriaga, Eugenio Massa. Existence, nonexistence and multiplicity of positive solutions for the poly-Laplacian and nonlinearities with zeros. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3831-3850. doi: 10.3934/dcds.2018166 [16] Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107 [17] Xiyou Cheng, Zhaosheng Feng, Zhitao Zhang. Multiplicity of positive solutions to nonlinear systems of Hammerstein integral equations with weighted functions. Communications on Pure & Applied Analysis, 2020, 19 (1) : 221-240. doi: 10.3934/cpaa.2020012 [18] Eun Kyoung Lee, R. Shivaji, Inbo Sim, Byungjae Son. Analysis of positive solutions for a class of semipositone p-Laplacian problems with nonlinear boundary conditions. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1139-1154. doi: 10.3934/cpaa.2019055 [19] Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach. Discrete & Continuous Dynamical Systems, 2008, 21 (4) : 1047-1069. doi: 10.3934/dcds.2008.21.1047 [20] Shao-Yuan Huang. Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1271-1294. doi: 10.3934/cpaa.2018061

2020 Impact Factor: 1.916