Let $\alpha \geq 1$. Suppose that for each $c\geq c_0>0$ there exists a point $\xi (c) \in ]0,1[$ s.t. the BVP:
$\begin{cases} [x^\alpha u^\prime (x)]^\prime +c\ u(x)=0 &\text{, in } ]\xi(c),1[ \\ u(\xi(c))=1,\ u(1)=0 \end{cases}$
has a positive, strictly decreasing, convex solution in $[\xi (c),1]$ with $u^\prime (\xi (c))=-c\ \xi^{1-\alpha}(c)$; moreover, assume that $\xi (c_0)=0$ and $\xi(\cdot)$ is strictly increasing in $[c_0,+\infty[$ and $\displaystyle \lim_{c\to c_0} \xi(c)=0$.
The question is:
Is it possible to prove an estimate of the type:
$\tag{1} \xi(c) \leq K\ (c-c_0)^\beta$
with $K,\beta >0$ suitable constants?
N.B.: (1) says that the point $\xi(c)$ cannot approach $\xi (c_0)=0$ too slowly when $c$ approaches $c_0$.
AFAIK, an estimate of type (1) holds in the case $\alpha=1$ with $\beta=1/2$, and it can be recovered using the power series expansion of Bessel functions. So I was wondering if there's an analogous result for $\alpha \neq 1$ and, in the positive case, if there are references where I can read it.