Bubbling along boundary geodesics near the second critical exponent

Abstract

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 Δ_u_ + up = 0, u > 0 under zero Dirichlet boundary conditions, in a domain Ω in ℝ_n_ 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|2 converges weakly to a Dirac measure on Γ as ε → 0+ exists, provided that Γ is non-degenerate in the sense of second variations of length and ε remains away from certain explicit discrete set of values for which a resonance phenomenon takes place.