### Abstract

We study Trudinger type inequalities in R^{N} and their best exponents αN. We show for α ε (0, αN), αN = Nω_{N-1}^{1/(N-1)} (ω_{N-1} is the surface area of the unit sphere in R^{N}), there exists a constant Cα > 0 such that (Equation Presented) for all u 6 ε W^{1,N}(R^{N}) \ {0}. Here Φ_{N}(ε) is defined by (Equation Presented) It is also shown that (*) with α ≥ α_{N} is false, which is different from the usual Trudinger's inequalities in bounded domains.

Original language | English |
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Pages (from-to) | 2051-2057 |

Number of pages | 7 |

Journal | Proceedings of the American Mathematical Society |

Volume | 128 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2000 Jan 1 |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

Adachi, S., & Tanaka, K. (2000). Trudinger type inequalities in R

^{N}and their best exponents.*Proceedings of the American Mathematical Society*,*128*(7), 2051-2057. https://doi.org/10.1090/s0002-9939-99-05180-1