Uniqueness of positive solutions for a singular nonlinear eigenvalue problem when a parameter is large

Abstract

We study positive solutions to singular boundary value problems of the form \begin{equation*} \begin{cases} -u''(t) = \lambda h(t) \frac{f(u(t))}{u(t)^\beta};~~(0,1)\\ u(0)=0=u(1), \end{cases} \end{equation*} where $\lambda >0$ is a parameter, $\beta \in (0,1), f:[0, \infty) \rightarrow (0,\infty)$ is a $C^1$ function, $\frac{f(s)}{s^\beta}$ is decreasing for $s \gg 1, h:(0,1) \rightarrow (0,\infty)$ is a continuous function and there exist $C_i>0,\alpha_i \in (0,1), i=1,2 $ such that $ h(t) \leq \frac{C_1}{t^{\alpha_1}};~ t \approx 0$ and $h(t) \leq \frac{C_2}{(1-t)^{\alpha_2}};~ t \approx 1$. We establish the uniqueness of positive solutions for $\lambda \gg 1,$ when $\alpha_i +\beta <1, i=1,2.$