Interaction between the Geometry of the Boundary and Positive Solutions of a Semilinear Neumann Problem with Critical Nonlinearity

We consider the problem: −Δu + λu = un + 2)(n − 2, u > 0 in Ω, ∂u/∂v = 0 on ∂Ω, where Ω is a bounded smooth domain in Rn (n ≥ 3). We show that, for λ large, least-energy solutions of the above problem have a unique maximum point Pλ on ∂Ω and the limit points of Pλ, as λ → ∞ are contained in the set of the points of maximum mean curvature. We also prove that, if ∂Ω has k peaks then the equation has at least k solutions for λ large.