Article Contents
Article Contents

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

• 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.

Mathematics Subject Classification: Primary: 34B15; Secondary: 34B18.

 Citation:

• Figure 1.  Graphs of bifurcation curves $S_{L}$ of (1)

Figure 2.  Graphs of $f(u)$ on $[0, \infty )$. (i) $q>p\geq 0$. (ii) $p>q\geq 0.$

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