Infinitely many turning points for an elliptic problem with a singular non-linearity

We consider the problem – u = |x|/(1 – u)^p inB, u = 0 on B, 0 < u < 1 in B, where 0, p 1 and Bis the unit ball in ^N (N 2). We show that there exists a _*> 0 such that for < _*, the minimizer is the only positiveradial solution. Furthermore, if , then the branch of positive radial solutions must undergoinfinitely many turning points as the maximums of the radialsolutions on the branch converge to 1. This solves ConjectureB in [N. Ghoussoub and Y. Gun, SIAM J. Math. Anal. 38 (2007)1423–1449]. The key ingredient is the use of monotonicityformula.