On S-shaped and reversed S-shaped bifurcation curves for singular problems

Abstract

We analyze the positive solutions to the singular boundary value problem \begin{equation*}\begin{cases} -(|u'|^{p-2}u)'=\lambda \frac{g(u)}{u^\beta}; (0,1),\\ u(0)=0=u(1), \end{cases}\end{equation*} where $p>1,\beta\in(0,1),\lambda>0$ and $g:[0,\infty) \rightarrow\mathbb{R}$ is a $C^1$ function. In particular, we discuss examples when $g(0)>0$ and when $g(0)<0$ that lead to $S$-shaped and reversed $S$-shaped bifurcation curves, respectively.

Keywords

• bifurcation