## Differential and Integral Equations

- Differential Integral Equations
- Volume 29, Number 7/8 (2016), 631-664.

### Classification and evolution of bifurcation curves for a one-dimensional prescribed mean curvature problem

Yan-Hsiou Cheng, Kuo-Chih Hung, and Shin-Hwa Wang

#### Abstract

We study the classification and evolution of bifurcation curves of positive solutions $u\in C^{2}(-L,L)\cap C[-L,L]$ for the one-dimensional prescribed mean curvature problem \begin{equation*} \left\{ \begin{array}{l} - \Big ( \dfrac{u^{\prime }(x)}{\sqrt{1+ ( {u^{\prime }(x)} ) ^{2}}} \Big ) ^{\prime }=\lambda [ \exp (\frac{au}{a+u})-1 ] , \ \ -L < x < L,\\ u(-L)=u(L)=0, \end{array} \right. \end{equation*} where $\lambda >0$ is a bifurcation parameter, and $L,a>0$ are two evolution parameters. We prove that, on $(\lambda , \Vert u \Vert _{\infty })$ -plane, the bifurcation curve of this problem is (i) $\supset $-shaped when $ 0 < a < 2$; (ii) $\supset $-shaped or monotone decreasing when $a=2$; (iii) reversed $S$-like shaped or monotone decreasing when $a>2$. Moreover, for $ a>2,$ the bifurcation curve is reversed $S$-like shaped if $L>0$ is large enough and is monotone decreasing if $L>0$ is small enough.

#### Article information

**Source**

Differential Integral Equations, Volume 29, Number 7/8 (2016), 631-664.

**Dates**

First available in Project Euclid: 3 May 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1462298679

**Mathematical Reviews number (MathSciNet)**

MR3498871

**Zentralblatt MATH identifier**

06604489

**Subjects**

Primary: 34B18: Positive solutions of nonlinear boundary value problems 74G35: Multiplicity of solutions

#### Citation

Cheng, Yan-Hsiou; Hung, Kuo-Chih; Wang, Shin-Hwa. Classification and evolution of bifurcation curves for a one-dimensional prescribed mean curvature problem. Differential Integral Equations 29 (2016), no. 7/8, 631--664. https://projecteuclid.org/euclid.die/1462298679