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

Abstract

In this paper, we study the global bifurcation curves and the exact multiplicity of positive solutions for the one-dimensional Minkowski-curvature problem \begin{document}$ \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. $\end{document} where \begin{document}$ p, q\geq 0 $\end{document} , \begin{document}$ p\neq q $\end{document} , \begin{document}$ \lambda >0 $\end{document} is a bifurcation parameter and \begin{document}$ L>0 $\end{document} is an evolution parameter. We prove that the bifurcation curve is continuous and further classify its exact shape (either monotone increasing or \begin{document}$ \subset $\end{document} -shaped by \begin{document}$ p $\end{document} and \begin{document}$ q $\end{document} ). Moreover, we can achieve the exact multiplicity of positive solutions.