# American Institute of Mathematical Sciences

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:

show all references

##### References:
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 $\left( 0, \infty \right) \times (0, 2.01].$
 [1] Shao-Yuan Huang. Global bifurcation and exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with sign-changing nonlinearity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3267-3284. doi: 10.3934/cpaa.2019147 [2] Shao-Yuan Huang. Bifurcation diagrams of positive solutions for one-dimensional Minkowski-curvature problem and its applications. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3443-3462. doi: 10.3934/dcds.2019142 [3] Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159-169. doi: 10.3934/proc.2013.2013.159 [4] Tetsuya Ishiwata, Takeshi Ohtsuka. Evolution of a spiral-shaped polygonal curve by the crystalline curvature flow with a pinned tip. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5261-5295. doi: 10.3934/dcdsb.2019058 [5] Ruyun Ma, Man Xu. Connected components of positive solutions for a Dirichlet problem involving the mean curvature operator in Minkowski space. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2701-2718. doi: 10.3934/dcdsb.2018271 [6] Matthias Bergner, Lars Schäfer. Time-like surfaces of prescribed anisotropic mean curvature in Minkowski space. Conference Publications, 2011, 2011 (Special) : 155-162. doi: 10.3934/proc.2011.2011.155 [7] Hongjie Ju, Jian Lu, Huaiyu Jian. Translating solutions to mean curvature flow with a forcing term in Minkowski space. Communications on Pure & Applied Analysis, 2010, 9 (4) : 963-973. doi: 10.3934/cpaa.2010.9.963 [8] Yves Coudène, Barbara Schapira. Counterexamples in non-positive curvature. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1095-1106. doi: 10.3934/dcds.2011.30.1095 [9] Alejandro Allendes, Alexander Quaas. Multiplicity results for extremal operators through bifurcation. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 51-65. doi: 10.3934/dcds.2011.29.51 [10] Alessio Pomponio. Oscillating solutions for prescribed mean curvature equations: euclidean and lorentz-minkowski cases. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 3899-3911. doi: 10.3934/dcds.2018169 [11] Tatsuki Mori, Kousuke Kuto, Tohru Tsujikawa, Shoji Yotsutani. Exact multiplicity of stationary limiting problems of a cell polarization model. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5627-5655. doi: 10.3934/dcds.2016047 [12] Junping Shi, Ratnasingham Shivaji. Exact multiplicity of solutions for classes of semipositone problems with concave-convex nonlinearity. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 559-571. doi: 10.3934/dcds.2001.7.559 [13] Jifeng Chu, Delia Ionescu-Kruse, Yanjuan Yang. Exact solution and instability for geophysical waves at arbitrary latitude. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4399-4414. doi: 10.3934/dcds.2019178 [14] Josef DiblÍk, Rigoberto Medina. Exact asymptotics of positive solutions to Dickman equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 101-121. doi: 10.3934/dcdsb.2018007 [15] Klaus-Jochen Engel, Marjeta Kramar FijavŽ. Exact and positive controllability of boundary control systems. Networks & Heterogeneous Media, 2017, 12 (2) : 319-337. doi: 10.3934/nhm.2017014 [16] Tzung-shin Yeh. S-shaped and broken s-shaped bifurcation curves for a multiparameter diffusive logistic problem with holling type-Ⅲ functional response. Communications on Pure & Applied Analysis, 2017, 16 (2) : 645-670. doi: 10.3934/cpaa.2017032 [17] Mitsunori Nara, Masaharu Taniguchi. Convergence to V-shaped fronts in curvature flows for spatially non-decaying initial perturbations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 137-156. doi: 10.3934/dcds.2006.16.137 [18] 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 [19] Yi-hsin Cheng, Tsung-Fang Wu. Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2457-2473. doi: 10.3934/cpaa.2016044 [20] 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

2018 Impact Factor: 0.925