# American Institute of Mathematical Sciences

2018, 17(3): 1271-1294. doi: 10.3934/cpaa.2018061

## Exact multiplicity and bifurcation curves of positive solutions of a one-dimensional Minkowski-curvature problem and its application

 Center for General Education, National Formosa University, Yunlin 632, Taiwan

* Corresponding author: Shao-Yuan Huang

Received  July 2016 Revised  May 2017 Published  January 2018

In this paper, we discuss exact multiplicity and bifurcation curves of positive solutions of the one-dimensional Minkowski-curvature problem
 $\begin{equation*}\left\{\begin{array}{l}\left[ -u^{\prime }/\sqrt{1-u^{\prime 2}}\right] ^{\prime }=\lambda f(u),\,\,\,\,\,\,-L < x < L, \\u(-L)=u(L)=0,\end{array}\right.\end{equation*}$
where $λ >0$ is a bifurcation parameter, $L>0$ is an evolution parameter,
 $f∈ C[0, ∞)\cap C^{2}(0, ∞),$
 $f(u)>0$
for
 $u>0$
, and
 $f^{\prime \prime }(u)$
is not sign-changing on
 $\left( 0,\infty \right)$
.We find that if
 $f^{\prime \prime }(u)≤q 0$
for
 $u>0$
, the shapes of bifurcation curves are monotone increasing for $L>0$, and if
 $f^{\prime \prime }(u)>0$
for
 $u>0$
and
 $f(u)$
satisfies some suitable hypotheses, the shapes of bifurcation curves has three possibilities. Furthermore, we study, in the
 $(\lambda ,L,{\left\| u \right\|_\infty })$
-space, the shapes and structures of the bifurcation surfaces. Finally, we give an application for this problem with a nonlinear term
 $f(u) = u^{p}+u^{q}$
where
 $q≥p>0$
satisfy some conditions.
Citation: 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
##### References:
 [1] R. Bartnik and L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys., 87 (1982), 131-152. doi: 10.1007/BF01211061. [2] C. Bereanu, P. Jebelean and P. Torres, Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space, J. Funct. Anal., 264 (2013), 270-287. doi: 10.1016/j.jfa.2012.10.010. [3] C. Bereanu, P. Jebelean and P. Torres, Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space, J. Funct. Anal., 265 (2013), 644-659. doi: 10.1016/j.jfa.2013.04.006. [4] C. Corsato, Mathematical Analysis of Some Differential Models Involving the Euclidean or the Minkowski Mean Curvature Operator, Ph. D thesis, Universita degli studi di Trieste, 2013. [5] P. M. Cohn, Basic Algebra: Groups, Gings and Fields, Springer-Verlag, London, 2003. doi: 10.1007/978-0-85729-428-9. [6] I. Coelho, C. Corsato, F. Obersnel and P. Omari, Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation, Adv. Nonlinear Stud., 12 (2012), 621-638. [7] I. Coelho, C. Corsato and S. Rivetti, Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball, Topol. Methods Nonlinear Anal., 44 (2014), 23-39. doi: 10.12775/TMNA.2014.034. [8] P. Clément, R. Manásevich and E. Mitidieri, On a modified capillary equation, J. Differential Equations, 124 (1996), 343-358. doi: 10.1006/jdeq.1996.0013. [9] C. Corsato, F. Obersnel, P. Omari and S. Rivetti, On the lower and upper solution method for the prescribed mean curvature equation in minkowski space, Discrete Contin. Dyn. Syst. 2013, Dynamical Systems, Differential Equations and Applications. 9th AIMS Conference. Suppl., 159-169. [10] C. Corsato, F. Obersnel, P. Omari and S. Rivetti, Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space, J. Math. Anal. Appl., 405 (2013), 227-239. doi: 10.1016/j.jmaa.2013.04.003. [11] C. V. Coffman and W. K. Ziemer, A prescribed mean curvature problem on domains without radial symmetry, SIAM J. Math. Anal., 22 (1991), 982-990. doi: 10.1137/0522063. [12] J. Forde and P. Nelson, Applications of Sturm sequences to bifurcation analysis of delay differential equation models, J. Math. Anal. Appl., 300 (2004), 273-284. doi: 10.1016/j.jmaa.2004.02.063. [13] S.-Y. Huang and S.-H. Wang, On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion, Discrete Contin. Dyn. Syst., 35 (2015), 4839-4858. doi: 10.3934/dcds.2015.35.4839. [14] S. -Y. Huang and S. -H. Wang, A proof and some verifications by symbolic manipulator Maple 16 (2015). Available form http://mx.nthu.edu.tw/~sy-huang/Math/proofs . [15] 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. doi: 10.1016/j.jde.2014.06.013. [16] J. Mawhin, Nonlinear boundary value problems involving the extrinsic mean curvature operator, Math. Bohem., 139 (2014), 299-313. [17] A. Prestel and C. N. Delzell, Positive Polynomials: From Hilbert's 17th Problem to Real Algebra, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-662-04648-7. [18] J. Serrin, Positive Solutions of a Prescribed Mean Curvature Problem, in: Calculus of Variations and Differential Equations, in: Lecture Notes in Math., Springer-Verlag, New York, 1988. doi: 10.1007/BFb0082900. [19] M. Zhang and J. Deng, Number of zeros of interval polynomials, J. Comput. Appl. Math., 237 (2013), 102-110. doi: 10.1016/j.cam.2012.07.011. [20] 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. doi: 10.1016/j.jmaa.2012.05.053.

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##### References:
 [1] R. Bartnik and L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys., 87 (1982), 131-152. doi: 10.1007/BF01211061. [2] C. Bereanu, P. Jebelean and P. Torres, Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space, J. Funct. Anal., 264 (2013), 270-287. doi: 10.1016/j.jfa.2012.10.010. [3] C. Bereanu, P. Jebelean and P. Torres, Multiple positive radial solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space, J. Funct. Anal., 265 (2013), 644-659. doi: 10.1016/j.jfa.2013.04.006. [4] C. Corsato, Mathematical Analysis of Some Differential Models Involving the Euclidean or the Minkowski Mean Curvature Operator, Ph. D thesis, Universita degli studi di Trieste, 2013. [5] P. M. Cohn, Basic Algebra: Groups, Gings and Fields, Springer-Verlag, London, 2003. doi: 10.1007/978-0-85729-428-9. [6] I. Coelho, C. Corsato, F. Obersnel and P. Omari, Positive solutions of the Dirichlet problem for the one-dimensional Minkowski-curvature equation, Adv. Nonlinear Stud., 12 (2012), 621-638. [7] I. Coelho, C. Corsato and S. Rivetti, Positive radial solutions of the Dirichlet problem for the Minkowski-curvature equation in a ball, Topol. Methods Nonlinear Anal., 44 (2014), 23-39. doi: 10.12775/TMNA.2014.034. [8] P. Clément, R. Manásevich and E. Mitidieri, On a modified capillary equation, J. Differential Equations, 124 (1996), 343-358. doi: 10.1006/jdeq.1996.0013. [9] C. Corsato, F. Obersnel, P. Omari and S. Rivetti, On the lower and upper solution method for the prescribed mean curvature equation in minkowski space, Discrete Contin. Dyn. Syst. 2013, Dynamical Systems, Differential Equations and Applications. 9th AIMS Conference. Suppl., 159-169. [10] C. Corsato, F. Obersnel, P. Omari and S. Rivetti, Positive solutions of the Dirichlet problem for the prescribed mean curvature equation in Minkowski space, J. Math. Anal. Appl., 405 (2013), 227-239. doi: 10.1016/j.jmaa.2013.04.003. [11] C. V. Coffman and W. K. Ziemer, A prescribed mean curvature problem on domains without radial symmetry, SIAM J. Math. Anal., 22 (1991), 982-990. doi: 10.1137/0522063. [12] J. Forde and P. Nelson, Applications of Sturm sequences to bifurcation analysis of delay differential equation models, J. Math. Anal. Appl., 300 (2004), 273-284. doi: 10.1016/j.jmaa.2004.02.063. [13] S.-Y. Huang and S.-H. Wang, On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion, Discrete Contin. Dyn. Syst., 35 (2015), 4839-4858. doi: 10.3934/dcds.2015.35.4839. [14] S. -Y. Huang and S. -H. Wang, A proof and some verifications by symbolic manipulator Maple 16 (2015). Available form http://mx.nthu.edu.tw/~sy-huang/Math/proofs . [15] 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. doi: 10.1016/j.jde.2014.06.013. [16] J. Mawhin, Nonlinear boundary value problems involving the extrinsic mean curvature operator, Math. Bohem., 139 (2014), 299-313. [17] A. Prestel and C. N. Delzell, Positive Polynomials: From Hilbert's 17th Problem to Real Algebra, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-662-04648-7. [18] J. Serrin, Positive Solutions of a Prescribed Mean Curvature Problem, in: Calculus of Variations and Differential Equations, in: Lecture Notes in Math., Springer-Verlag, New York, 1988. doi: 10.1007/BFb0082900. [19] M. Zhang and J. Deng, Number of zeros of interval polynomials, J. Comput. Appl. Math., 237 (2013), 102-110. doi: 10.1016/j.cam.2012.07.011. [20] 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. doi: 10.1016/j.jmaa.2012.05.053.
Graphs of bifurcation curves $S_{L}$ of (1). (ⅰ) hypotheses of Theorem 2.1 (ⅰ) hold. (ⅱ) hypotheses of Theorem 2.1(ⅱ) hold. (ⅲ) hypotheses of Theorem 2.1(ⅲ) hold. (ⅳ) hypotheses of Theorem 2.1(ⅳ) hold. (ⅴ) hypotheses of Theorem 2.2 hold.
Graphs of bifurcation surface $\Gamma$ of (1). (ⅰ) hypotheses of Theorem 2.1(ⅰ) hold. (ⅱ) hypotheses of Theorem 2.1(ⅱ) hold. (ⅲ) hypotheses of Theorem 2.1 (ⅲ) hold. (ⅳ) hypotheses of Theorem 2.1(ⅳ) hold. (ⅴ) hypotheses of Theorem 2.2 hold.
Graphs of bifurcation set $\tilde{\Sigma}$ and $\bar{\Sigma}\equiv \{(\lambda, \frac{\pi }{2\sqrt{2\lambda \eta }}):\lambda >0\}$. (ⅰ) hypotheses of Theorem 2.1(ⅳ) hold. (ⅱ) hypotheses of Theorem 2.1(ⅲ) hold. (ⅲ) hypotheses of Theorem 2.2 hold.
Numerical simulations of bifurcation surfaces $\Gamma$ of ( 4). (ⅰ) $0<p<q\leq 1$ or $0<p = q<1.$ (ⅱ) $p = q = 1$. (ⅲ) $1 = p<q\leq 2$. (ⅳ) $1<p\leq q\leq \left( 2+\sqrt{3}\right) p-1-\sqrt{3}.$ (ⅴ) $p = 1,$ $q = 3,$ and $\hat{\lambda} = 2.$ The red curve $S_{\hat{L}}$ is monotone increasing.
The conjecture of global bifurcation curves $S_{L}$ for ( 4).
Graphs of $T_{\lambda }(\alpha )$ on $\left( 0, \infty \right)$. (ⅰ) ((C5) and (H)), or ((C6) and (H)), or ((C7) and (H)) holds. (ⅱ) (C4) and (H) hold. (ⅲ) (C9) holds and there exists $\check{ \lambda}>\hat{\lambda}$ such that (H) holds under $0< \lambda \leq \check{\lambda}$ where $\hat{\lambda}$ is defined in Theorem 2.2.
The sets $\Theta _{1}, \Theta _{2}, ..., \Theta _{6}$ in $\left( 0, \infty \right) \times (0, 2.01].$
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