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

We consider the problem -Delta u = lambda|x|(alpha)/(1 - u)(P) in B, u = 0 on delta B, 0 < u < 1 in B, where alpha >= 0, p >= 1 and B is the unit ball in R-N (N >= 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 < N < 2 + (2 (2 + alpha) (p + 1)) (p + root p2 + 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 (2007) 1423-1449]. The key ingredient is the use of monotonicity formula.