Exact boundary behavior of the unique positive solution for singular second-order differential equations

Abstract

In this paper, we give the exact asymptotic behavior of the unique positive solution to the following singular boundary value problem \begin{equation*} \begin{cases} -\frac{1}{A}(Au^{\prime })^{\prime }=p(x)g(u),\quad x\in (0,1), \\ u>0,\quad \text{in }(0,1), \\ \lim_{x\rightarrow 0^{+}}(Au^{\prime })(x)=0,\quad u(1)=0, \end{cases} \end{equation*} where $A$ is a continuous function on $[0,1),$ positive and differentiable on $(0,1)$ such that $\frac{1}{A}$ is integrable in a neighborhood of $1,$ $g\in C^{1}((0,\infty ),(0,\infty ))$ is nonincreasing on $(0,\infty )$ with $\lim_{t\rightarrow 0} g^{\prime }(t)\int_{0}^{t}\frac{1}{g(s)}\,ds=-C_{g}\leq 0$ and $p$ is a nonnegative continuous function in $(0,1)$ satisfying \begin{equation*} 0<p_{1}=\liminf_{x\rightarrow 1}\frac{p(x)}{h(1-x)}\leq \limsup_{x\rightarrow 1}\frac{p(x)}{h(1-x)}=p_{2}<\infty , \end{equation*} where $h(t)=ct^{-\lambda }\exp (\int_{t}^{\eta}\frac{z(s)}{s}\,ds),$ $\lambda \leq 2,$ $c>0$ and $z$ is continuous on $[0,\eta ]$ for some $\eta >1$ such that $z(0)=0.