## 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^{+}, 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.

Original language | English |
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Pages (from-to) | 1553-1605 |

Number of pages | 53 |

Journal | Journal of the European Mathematical Society |

Volume | 12 |

Issue number | 6 |

DOIs | |

Publication status | Published - 28 Sep 2010 |

## Keywords

- Blowing-up solution
- Critical Sobolev exponent
- Nondegenerate geodesic

## ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics