### 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 |
---|---|

Pages (from-to) | 2051-2057 |

Number of pages | 7 |

Journal | Proceedings of the American Mathematical Society |

Volume | 128 |

Issue number | 7 |

Publication status | Published - 2000 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

^{N}and their best exponents.

*Proceedings of the American Mathematical Society*,

*128*(7), 2051-2057.

**Trudinger type inequalities in R ^{N} and their best exponents.** / Adachi, Shinji; Tanaka, Kazunaga.

Research output: Contribution to journal › Article

^{N}and their best exponents',

*Proceedings of the American Mathematical Society*, vol. 128, no. 7, pp. 2051-2057.

^{N}and their best exponents. Proceedings of the American Mathematical Society. 2000;128(7):2051-2057.

}

TY - JOUR

T1 - Trudinger type inequalities in RN and their best exponents

AU - Adachi, Shinji

AU - Tanaka, Kazunaga

PY - 2000

Y1 - 2000

N2 - We study Trudinger type inequalities in RN and their best exponents αN. We show for α ε (0, αN), αN = NωN-1 1/(N-1) (ωN-1 is the surface area of the unit sphere in RN), there exists a constant Cα > 0 such that (Equation Presented) for all u 6 ε W1,N(RN) \ {0}. Here ΦN(ε) is defined by (Equation Presented) It is also shown that (*) with α ≥ αN is false, which is different from the usual Trudinger's inequalities in bounded domains.

AB - We study Trudinger type inequalities in RN and their best exponents αN. We show for α ε (0, αN), αN = NωN-1 1/(N-1) (ωN-1 is the surface area of the unit sphere in RN), there exists a constant Cα > 0 such that (Equation Presented) for all u 6 ε W1,N(RN) \ {0}. Here ΦN(ε) is defined by (Equation Presented) It is also shown that (*) with α ≥ αN is false, which is different from the usual Trudinger's inequalities in bounded domains.

UR - http://www.scopus.com/inward/record.url?scp=23044521004&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=23044521004&partnerID=8YFLogxK

M3 - Article

VL - 128

SP - 2051

EP - 2057

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

SN - 0002-9939

IS - 7

ER -