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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

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##### 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 $\widetilde{H}(\rho,0).$
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