Summary: We consider the problem \(-\Delta u = \lambda |x|^{\alpha }/(1 - u)^{p}\) in \(B, u = 0\) on \(\partial B, 0 < u < 1\) in \(B\), where \(\alpha \geqslant 0, p \geqslant 1\) and \(B\) is the unit ball in \(\mathbb R^{N} (N \geqslant 2)\). We show that there exists a \(\lambda _{*} > 0\) such that for \(\lambda < \lambda _{*}\), the minimizer is the only positive radial solution. Furthermore, if \(2 \leqslant N < 2 + (2(2 + \alpha )/(p + 1))(p + \sqrt{p^{2} + p})\), then the branch of positive radial solutions must undergo infinitely many turning points as the maximums of the radial solutions on the branch converge to 1. This solves Conjecture B in [N. Ghoussoub and Y. Gun, SIAM J. Math. Anal. 38, No. 5, 1423–1449 (2007; Zbl 1174.35040)]. The key ingredient is the use of monotonicity formula.