### Abstract

We study an elliptic equation related to the Moser–Trudinger inequality on a compact Riemann surface (S,g), Δ
_{g}u+λ(ue
^{u
2
}−[Formula presented]∫Sue
^{u
2
}dv
_{g})=0,in S,∫Sudv
_{g}=0, where λ>0 is a small parameter, |S| is the area of S, Δ
_{g} is the Laplace–Beltrami operator and dv
_{g} is the area element. Given any integer k≥1, under general conditions on S we find a bubbling solution u
_{λ} which blows up at exactly k points in S, as λ→0. When S is a flat two-torus in rectangular form, we find that either seven or nine families of such solutions do exist for k=2. In particular, in any square flat two-torus actually nine families of bubbling solutions with two bubbling points do exist. If S is a Riemann surface with non-constant Robin's function then at least two bubbling solutions with k=1 exists.

Original language | English |
---|---|

Pages (from-to) | 2684-2739 |

Number of pages | 56 |

Journal | Journal of Functional Analysis |

Volume | 275 |

Issue number | 10 |

Early online date | 27 Aug 2018 |

DOIs | |

Publication status | Published - 15 Nov 2018 |

### Keywords

- Green's function
- Moser–Trudinger inequality

### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Functional Analysis*,

*275*(10), 2684-2739. https://doi.org/10.1016/j.jfa.2018.08.016