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On Lin–Ni's conjecture in convex domains
Author(s) -
Wang Liping,
Wei Juncheng,
Yan Shusen
Publication year - 2011
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdq051
Subject(s) - mathematics , uniqueness , conjecture , bounded function , neumann boundary condition , domain (mathematical analysis) , corollary , regular polygon , combinatorics , a priori estimate , pure mathematics , mathematical analysis , boundary (topology) , geometry
We consider the following non‐linear Neumann problem:{− Δ u + μ u = u ( N + 2 ) / ( N − 2 ) , u > 0in Ω ,∂ u ∂ n = 0on ∂ Ω ,where μ > 0, Ω is a bounded domain inR Nand n denotes the outward unit normal of ∂ Ω. Lin and Ni ( On the diffusion coefficient of a semilinear Neumann problem , Lecture Notes in Mathematics 1340 (Springer, Berlin, 1986) 160–174) conjectured that, for μ small, all solutions are constants . It has been shown in (J. Wei and X. Xu, ‘Uniqueness and a priori estimates for some nonlinear elliptic Neumann equations inR 3 ’, Pacific J. Math. 221 (2005) 159–165; M. Zhu, ‘Uniqueness results through a priori estimates, I. A three dimensional Neumann problem’, J. Differential Equations 154 (1999) 284–317) that this conjecture is true if Ω is convex and N = 3. The main result of this paper is that if N ⩾ 4, Ω is convex and satisfies some symmetric conditions, then, for any fixed μ , there are infinitely many positive solutions . As a corollary, the Lin–Ni's conjecture is false in some convex domains if N ⩾ 4.