## Abstract

In this paper, we study the existence and non-existence of maximizers for the Moser–Trudinger type inequalities in R ^{N} of the form DN,α(a,b):=supu∈W1,N(RN),‖∇u‖LN(RN)a+‖u‖LN(RN)b=1∫RNΦN(α|u|N′)dx.Here N≥2,N′=NN-1,a,b>0,α∈(0,αN] and ΦN(t):=et-∑j=0N-2tjj! where αN:=NωN-11/(N-1) and ω _{N} _{-} _{1} denotes the surface area of the unit ball in R ^{N} . We show the existence of the threshold α _{∗} = α _{∗} (a, b, N) ∈ [0 , α _{N} ] such that D _{N} _{,} _{α} (a, b) is not attained if α∈ (0 , α _{∗} ) and is attained if α∈ (α _{∗} , α _{N} ). We also provide the conditions on (a, b) in order that the inequality α _{∗} < α _{N} holds.

Original language | English |
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Pages (from-to) | 831-851 |

Number of pages | 21 |

Journal | Mathematische Annalen |

Volume | 373 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 2019 Feb 8 |

## ASJC Scopus subject areas

- Mathematics(all)