### Abstract

Let Ω be a bounded, smooth domain in ℝ ^{2}. We consider the functional, in the supercritical Trudinger-Moser regime, i. e. for ∫ _{Ω}{pipe}▼u{pipe} ^{2}dx > 4π. More precisely, we are looking for critical points of I(u) in the class of functions u ε H _{0} ^{1}(Ω) such that ∫ _{Ω}{pipe}▼u{pipe} ^{2}dx = 4π(1+α), for small α > 0. In particular, we prove the existence of 1-peak critical points of I(u) with ∫ _{Ω}{pipe}▼u{pipe} ^{2}dx = 4π(1+α) for any bounded domain Ω, 2-peak critical points with ∫ _{Ω}{pipe}▼u{pipe} ^{2}dx = 8π(1+α) for non-simply connected domains Ω, and k-peak critical points with ∫ _{Ω}{pipe}▼u{pipe} ^{2}dx = 4kπ(1+α) if Ω is an annulus.

Original language | English |
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Pages (from-to) | 543-576 |

Number of pages | 34 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 44 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 1 Jul 2012 |

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

## Cite this

*Calculus of Variations and Partial Differential Equations*,

*44*(3-4), 543-576. https://doi.org/10.1007/s00526-011-0444-5