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 $\Delta u+u^p=0$, $u>0$ under zero Dirichlet boundary conditions, in a domain $\Omega$ in $R^n$ with bounded, smooth boundary. Given $\Gamma$, a geodesic of the boundary with negative inner normal curvature we find that for $p=(n+1)/(n-3)-\epsilon$, there exists a solution $u_{\epsilon }$ such that $|\nabla u_{\epsilon }{|}^2$ converges weakly to a Dirac measure on $\Gamma$ as $\epsilon \to 0^{+}$ exists, provided that $\Gamma$ is non-degenerate in the sense of second variations of length and $\epsilon$ remains away from certain explicit discrete set of values for which a resonance phenomenon takes place.