On Lin-Ni's conjecture in convex domains

We consider the following non-linear Neumann problem: {-Δu + μu = u^(N+2)/(N-2), u > 0 in Ω, ∂u/∂n = 0 on ∂Ω, where μ > 0, Ω is a bounded domain in ℝ^N and 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 in ℝ³', 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.