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 + up = 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 uε 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. © European Mathematical Society 2010.