Global Bifurcation Diagrams for Liouville–Bratu–Gelfand Problem with Minkowski-Curvature Operator

Abstract

In this paper, we study global bifurcation diagrams and the exact multiplicity of positive solutions for Minkowski-curvature problem $$\begin{aligned} \left\{ \begin{array}{l} -\left( u^{\prime }/\sqrt{1-{u^{\prime }}^{2}}\right) ^{\prime }=\lambda \exp u,\text { in }\left( -L,L\right) , \\ u(-L)=u(L)=0, \end{array} \right. \end{aligned}$$ where $$\lambda >0$$ is a bifurcation parameter and $$L>0$$ is an evolution parameter. It can be viewed as a variant of the one-dimensional Liouville–Bratu–Gelfand problem. We prove that there exists $$L_{0}>0$$ such that the bifurcation curve $$S_{L}$$ is S-shaped for $$L>L_{0}$$ and is monotone increasing for $$0<L\le L_{0}$$ . We also study, in the $$\left( L,\lambda ,\left\| u\right\| _{\infty }\right) $$ -space, the shape and structure of the bifurcation surface. Finally, we will make a list which shows the different properties of bifurcation curves for Minkowski-curvature problem, corresponding semilinear problem and corresponding prescribed curvature problem.