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

We consider the problem − Δ u = λ | x | α /(1 − u ) p in B, u = 0 on ∂ B , 0 < u < 1 in B , where α ⩾ 0, p ⩾ 1 and B is the unit ball in ℝ N ( N ⩾ 2). We show that there exists a λ * > 0 such that for λ < λ * , the minimizer is the only positive radial solution. Furthermore, if 2 ⩽ N < 2 + ( 2 ( 2 + α ) / ( p + 1 ) ) ( p +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 (2007) 1423–1449]. The key ingredient is the use of monotonicity formula.