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

Zongming Guo, Juncheng Wei

• We consider the problem � u = |x|/(1 � u)p in B, u = 0 on B, 0 < u < 1 in B, where 0, p 1 and B is the unit ball in N (N 2). We show that there exists a * > 0 such that for < *, the minimizer is the only positive radial solution. Furthermore, if , 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.