Uniqueness results for a singular non-linear Sturm-Liouville equation (Q2828640)

This paper is devoted to the uniqueness of solutions to the problem NEWLINE\[NEWLINE\begin{aligned}-(x^{2\alpha}u')'&=\lambda u+u^p,\;x\in(0,1),\\ u(t)>0\;& \text{on}\;(0,1),\;\;u(1)=0.\end{aligned}NEWLINE\]NEWLINE The author shows that in the cases when \(0<\alpha<\frac{1}{2}\), \(\lambda\in \mathbb R\) and \(p>1\), and when \(\frac{1}{2}\leq\alpha<1,\) \(\lambda\in \mathbb R\) and \(1<p\leq\frac{3-2\alpha}{2\alpha-1}\) this problem has at most one \(C[0,1]\)-solution with the property \(x^{2\alpha-1}u'\in C[0,1];\) in fact, in the first case it has also the property \(\lim_{x\to0^+}u(x)=0.\)