### Abstract

Let Ω be an open bounded domain in R

^{n}with smooth boundary ∂Ω. We consider the equation ∆*u + u n−k+2 n−k−2−ε*= 0 in Ω, under zero Dirichlet boundary condition, where ε is a small positive parameter. We assume that there is a*k*-dimensional closed, embedded minimal sub-manifold*K*of ∂Ω, which is non-degenerate, and along which a certain weighted average of sectional curvatures of ∂Ω is negative. Under these assumptions, we prove existence of a sequence*ε = εj*and a solution*u*which concentrate along_{ε}*K*, as*ε*→ 0+, in the sense that |∇*u*|2 * S_{ε}*n−k 2 n−k δK*as*ε → 0*where δK stands for the Dirac measure supported on*K*and*Sn−k*is an explicit positive constant. This result generalizes the one obtained in [17], where the case*k*= 1 is considered.Original language | English |
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Pages (from-to) | 379-415 |

Number of pages | 37 |

Journal | Proceedings of the London Mathematical Society |

Volume | 118 |

Issue number | 2 |

Early online date | 2 Aug 2018 |

DOIs | |

Publication status | Published - 1 Feb 2019 |

### Fingerprint

### Keywords

- 35B40
- 35J10
- 35J61
- 58C15 (primary)

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

Deng, S., Mahmoudi, F., & Musso, M. (2019). Concentration at sub-manifolds for an elliptic Dirichlet problem near high critical exponents.

*Proceedings of the London Mathematical Society*,*118*(2), 379-415. https://doi.org/10.1112/plms.12183