    May  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:
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##### References:
  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  P. M. Cohn, Basic Algebra: Groups, Gings and Fields, Springer-Verlag, London, 2003. doi: 10.1007/978-0-85729-428-9.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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 . Google Scholar  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.  Google Scholar  J. Mawhin, Nonlinear boundary value problems involving the extrinsic mean curvature operator, Math. Bohem., 139 (2014), 299-313. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar 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. 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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