Bifurcation curves for the one-dimensional perturbed Gelfand problem with the Minkowski-curvature operator

Abstract

In this paper, we study the shapes of bifurcation curves of positive solutions for the one-dimensional perturbed Gelfand problem with the Minkowski-curvature operator$\{\begin{array}{c}-{\left(\frac{u^{\prime }\left(x\right)}{\sqrt{1-{\left(u^{\prime }\right(x\left)\right)}^2}}\right)}^{\prime }=\lambda \exp \left(\frac{au}{a+u}\right),-L<x<L,\\ u(-L)=u\left(L\right)=0,\end{array}$where $\lambda >0$ is a bifurcation parameter and $a,L>0$ are evolution parameters. We determine the shapes of the bifurcation curves for different positive values a and L.