Bubbling along boundary geodesics near the second critical exponent

The role of the second critical exponent p = (n + 1)/(n - 3), the Sobolev critical exponent in one dimension less, is investigated for the classical Lane-Emden-Fowler problem Δu + u^p = 0, u > 0 under zero Dirichlet boundary conditions, in a domain Ω in ℝⁿ with bounded, smooth boundary. Given Γ, a geodesic of the boundary with negative inner normal curvature we find that for p = (n + 1)/(n - 3) - ε, there exists a solution u_ε such that |∇u ^ε|²converges weakly to a Dirac measure on Γ as ε → 0⁺, provided that γ is nondegenerate in the sense of second variations of length and ε remains away from a certain explicit discrete set of values for which a resonance phenomenon takes place.