### Abstract

^{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 |
---|---|

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

*Proceedings of the London Mathematical Society*,

*118*(2), 379-415. https://doi.org/10.1112/plms.12183

**Concentration at sub-manifolds for an elliptic Dirichlet problem near high critical exponents.** / Deng, Shengbing; Mahmoudi, Fethi; Musso, Monica.

Research output: Contribution to journal › Article

*Proceedings of the London Mathematical Society*, vol. 118, no. 2, pp. 379-415. https://doi.org/10.1112/plms.12183

}

TY - JOUR

T1 - Concentration at sub-manifolds for an elliptic Dirichlet problem near high critical exponents

AU - Deng, Shengbing

AU - Mahmoudi, Fethi

AU - Musso, Monica

PY - 2019/2/1

Y1 - 2019/2/1

N2 - Let Ω be an open bounded domain in Rn 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 uε which concentrate along K, as ε → 0+, in the sense that |∇uε|2 * S 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.

AB - Let Ω be an open bounded domain in Rn 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 uε which concentrate along K, as ε → 0+, in the sense that |∇uε|2 * S 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.

KW - 35B40

KW - 35J10

KW - 35J61

KW - 58C15 (primary)

UR - http://www.scopus.com/inward/record.url?scp=85052630549&partnerID=8YFLogxK

U2 - 10.1112/plms.12183

DO - 10.1112/plms.12183

M3 - Article

VL - 118

SP - 379

EP - 415

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

SN - 0024-6115

IS - 2

ER -