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Infinitely many turning points for an elliptic problem with a singular non‐linearity
Author(s) -
Guo Zongming,
Wei Juncheng
Publication year - 2008
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdm121
Subject(s) - unit sphere , conjecture , mathematics , monotonic function , combinatorics , linearity , ball (mathematics) , mathematical analysis , pure mathematics , physics , quantum mechanics
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.