On Lin–Ni's conjecture in convex domains

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.