Existence and boundary behavior of positive solutions for a Sturm-Liouville problem

Abstract

In this paper, we discuss existence, uniqueness and boundary behavior of a positive solution to the following nonlinear Sturm-Liouville problem $\begin{aligned}&\frac{1}{A}(Au^{\prime })^{\prime }+a(t)u^{\sigma}=0;;\text{in};(0,1),\ &\lim\limits_{t\to 0}Au^{\prime}(t)=0,\quad u(1)=0,\end{aligned}$ where $\sigma \lt 1$, $A$ is a positive differentiable function on $(0,1)$ and $a$ is a positive measurable function in $(0,1)$ satisfying some appropriate assumptions related to the Karamata class. Our main result is obtained by means of fixed point methods combined with Karamata regular variation theory.