Classification and evolution of bifurcation curves for a one-dimensional prescribed mean curvature problem

Abstract

We study the classification and evolution of bifurcation curves of positive solutions $u\in C^{2}(-L,L)\cap C[-L,L]$ for the one-dimensional prescribed mean curvature problem \begin{equation*} \left\{ \begin{array}{l} - \Big ( \dfrac{u^{\prime }(x)}{\sqrt{1+ ( {u^{\prime }(x)} ) ^{2}}} \Big ) ^{\prime }=\lambda [ \exp (\frac{au}{a+u})-1 ] , \ \ -L < x < L,\\ u(-L)=u(L)=0, \end{array} \right. \end{equation*} where $\lambda >0$ is a bifurcation parameter, and $L,a>0$ are two evolution parameters. We prove that, on $(\lambda , \Vert u \Vert _{\infty })$ -plane, the bifurcation curve of this problem is (i) $\supset $-shaped when $ 0 < a < 2$; (ii) $\supset $-shaped or monotone decreasing when $a=2$; (iii) reversed $S$-like shaped or monotone decreasing when $a>2$. Moreover, for $ a>2,$ the bifurcation curve is reversed $S$-like shaped if $L>0$ is large enough and is monotone decreasing if $L>0$ is small enough.