Interaction between the geometry of the boundary and positive solutions of a semilinear neumann problem with critical nonlinearity

Abstract

We consider the problem: -Δu + λu = u^{n + 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 P_λ on ∂Ω and the limit points of P_λ, 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.