# American Institute of Mathematical Sciences

• Previous Article
Translating solutions of non-parametric mean curvature flows with capillary-type boundary value problems
• CPAA Home
• This Issue
• Next Article
The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line
November  2019, 18(6): 3267-3284. doi: 10.3934/cpaa.2019147

## Global bifurcation and exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem with sign-changing nonlinearity

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

Received  July 2018 Revised  December 2018 Published  May 2019

In this paper, we study the global bifurcation curves and the exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem
 \left\{ \begin{array}{*{35}{l}} \begin{align} & -{{\left( {{u}^{\prime }}/\sqrt{1-{{u}^{\prime }}^{2}} \right)}^{\prime }}=\lambda \left( {{u}^{p}}-{{u}^{q}} \right),\ \ \ \text{in}\left( {-L},{L} \right),\ \\ & u(-L)=u(L)=0, \\ \end{align} \\\end{array} \right.
where
 $p, q\geq 0$
,
 $p\neq q$
,
 $\lambda >0$
is a bifurcation parameter and
 $L>0$
is an evolution parameter. We prove that the bifurcation curve is continuous and further classify its exact shape (either monotone increasing or
 $\subset$
-shaped by
 $p$
and
 $q$
). Moreover, we can achieve the exact multiplicity of positive solutions.
Citation: 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
##### References:

show all references

##### References:
Graphs of bifurcation curves $S_{L}$ of (1)
Graphs of $f(u)$ on $[0, \infty )$. (i) $q>p\geq 0$. (ii) $p>q\geq 0.$