Boundary effect for an elliptic neumann problem with critical nonlinearity

Abstract

We are interested in elliptic problems with critical nonlinearity and Neumann boundary conditions, namely (P_μ) : -Δu + μu = u^(n+2)/(n-2), u>0 in Ω, ∂u/∂ν = 0 on ∂Ω — where Ω is a smooth bounded domain in ℝ^n, n≥3, and μ is a strictly positive parameter. We show, for n≥7, and u a small energy solution to (P_μ), that u concentrates as μ goes to infinity at a point of the boundary such that the mean curvature H is positive, and critical if it is strictly positive. Conversely we show, for n≥5, and α>0 a critical value of H inducing a difference of topology between the level sets of H, that there exists for μ large enough a solution to (P_μ) which concentrates at a point y of the boundary such that H(y) = α and H'(y) = 0. Lastly, if n≥6 and y_1, … , y_k are k distinct critical points of H, there exists for μ large enough a solution to (P_μ) which concentrates at each of the points y_i, 1≤i≤k.