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Journal of the London Mathematical Society Advance Access published online on March 20, 2008
Journal of the London Mathematical Society, doi:10.1112/jlms/jdm121
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© 2008 London Mathematical Society

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

Zongming Guo

Department of Mathematics
Henan Normal University
Xinxiang 453002
PR China
guozm@public.xxptt.ha.cn

Juncheng Wei

Department of Mathematics
The Chinese University of Hong Kong
Shatin
Hong Kong

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} ≥ 0, p ≥ 1 and B is the unit ball in RN (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 Formula , 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.

2000 Mathematics Subject Classification 35B45 (primary), 35J40 (secondary).

Received November 10, 2006; revised September 14, 2007;
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