# Bubbling along boundary geodesics near the second critical exponent

### Manuel del Pino

Universidad de Chile, Santiago, Chile### Monica Musso

Ponificia Universidad Catolica de Chile, Santiago, Chile### Frank Pacard

École Polytechnique, Palaiseau, France

## 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 $Δu+u_{p}=0$, $u>0$ 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.

## Cite this article

Manuel del Pino, Monica Musso, Frank Pacard, Bubbling along boundary geodesics near the second critical exponent. J. Eur. Math. Soc. 12 (2010), no. 6, pp. 1553–1605

DOI 10.4171/JEMS/241