| Summary: | In this article, we give an exact behavior at infinity of the unique
solution to the following singular boundary value problem
$$\displaylines{
-\frac{1}{A}(Au')'=q(t)g(u), \quad t \in (0,\infty), \cr
u>0, \quad \lim_{t\to 0}A(t)u'(t)=0, \quad \lim_{t\to \infty}u(t)=0.
}$$
Here A is a nonnegative continuous function on $[0,\infty)$, positive
and differentiable on $(0,\infty)$ such that
$$
\lim_{t\to \infty}\frac{tA'(t)}{A(t)}=\alpha>1, \quad
g \in C^1((0,\infty),(0,\infty))
$$
is non-increasing on $(0,\infty)$ with
$\lim_{t\to 0}g'(t)\int_0^t\frac{ds}{g(s)}=-C_g\leq 0$ and the function q
is a nonnegative continuous, satisfying
$$
0<a_1=\liminf_{t\to \infty}\frac{q(t)}{h(t)}
\leq \limsup_{t\to \infty}\frac{q(t)}{h(t)}=a_2<\infty,
$$
where $h(t)=c t^{-\lambda}\exp (\int_{1}^{t }\frac{y(s)}{s}ds)$,
$\lambda \geq 2$, $c>0$ and y is continuous on $[ 1,\infty)$
such that $\lim_{t\to \infty}y(t)=0$.
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