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

Shao-Yuan Huang

doi:10.3934/cpaa.2019147

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2019, Volume 18, Issue 6: 3267-3284. Doi: 10.3934/cpaa.2019147

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

Shao-Yuan Huang

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

Received: July 2018

Revised: December 2018

Published: May 2019

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Abstract

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.

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

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

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

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