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Trudinger type inequalities in $\mathbf{R}^N$
and their best exponents


Authors: Shinji Adachi and Kazunaga Tanaka
Journal: Proc. Amer. Math. Soc. 128 (2000), 2051-2057
MSC (1991): Primary 46E35, 26D10
DOI: https://doi.org/10.1090/S0002-9939-99-05180-1
Published electronically: November 1, 1999
MathSciNet review: 1646323
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Abstract: We study Trudinger type inequalities in ${\mathbf{R}}^{N}$ and their best exponents $\alpha _{N}$. We show for $\alpha \in (0,\alpha _{N})$, $\alpha _{N}=N\omega _{N-1}^{1/(N-1)}$ ($\omega _{N-1}$ is the surface area of the unit sphere in ${\mathbf{R}}^{N}$), there exists a constant $C_{\alpha }>0$ such that

\begin{equation*}\tag{$*$} \int _{\mathbf{R} ^{N}} \Phi _{N}\left (\alpha \left ( \frac{\left |u(x)\right | }{\|\nabla u\| _{L^{N}(\mathbf{R} ^{N})}} \right )^{\frac{N}{N-1}}\right )\, dx \leq C_{\alpha } \frac {\|u\|_{L^{N}(\mathbf{R} ^{N})} ^{N}}{\|\nabla u\|_{L^{N}(\mathbf{R} ^{N})}^{N}} \end{equation*}

for all $u \in W^{1,N} (\mathbf{R} ^{N})\setminus \{ 0\}$. Here $\Phi _{N}(\xi )$ is defined by

\begin{equation*}\Phi _{N}(\xi ) = \exp (\xi ) - \sum _{j=0}^{N-2} {\frac{1}{j!}}\xi ^{j}. \end{equation*}

It is also shown that $(*)$ with $\alpha \geq \alpha _{N}$ is false, which is different from the usual Trudinger's inequalities in bounded domains.


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  • [1] D. R. Adams, A sharp inequality of J. Moser for higher order derivatives, Ann. of Math. 128 (1988), 385-398. MR 89i:46034
  • [2] R. A. Adams, Sobolev Spaces, Academic Press, New York 1975. MR 56:9247
  • [3] L. Carleson and S.-Y. A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sc. Math. 110 (1986), 113-127. MR 88f:46070
  • [4] D. E. Edmunds and A. A. Ilyin, Asymptotically sharp multiplicative inequalities, Bull. London Math. Soc. 27 (1995), 71-74. MR 97b:26015
  • [5] M. Flucher, Extremal functions for the Trudinger-Moser inequality in 2 dimensions, Comm. Math. Helvetici 67 (1992), 471-497. MR 93k:58073
  • [6] H. Kozono, T. Ogawa and H. Sohr, Asymptotic behaviour in $L^{r}$ for weak solutions of the Navier-Stokes equations in exterior domains, Manuscripta Math. 74 (1992), 253-275. MR 92k:35220
  • [7] J. B. McLeod and L. A. Peletier, Observations on Moser's inequality, Arch. Rat. Mech. Anal. 106 (1989), 261-285. MR 90d:26029
  • [8] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1979), 1077-1092. MR 46:662
  • [9] T. Ogawa, A proof of Trudinger's inequality and its application to nonlinear
    Schrödinger equations, Nonlinear Anal. 14 (1990), 765-769. MR 91d:35203
  • [10] T. Ogawa and T. Ozawa, Trudinger type inequalities and uniqueness of weak solutions for the nonlinear Schrödinger mixed problem, J. Math. Anal. Appl. 155 (1991), 531-540. MR 92a:35147
  • [11] T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal. 127 (1995) 259-269. MR 96c:46039
  • [12] R. S. Strichartz, A note on Trudinger's extension of Sobolev's inequalities, Indiana Univ. Math. J. 21 (1972), 841-842. MR 45:2466
  • [13] M. Struwe, Critical points of embeddings of $H^{1,n}_{0}$ into Orlicz spaces, Ann. Inst. Henri Poincaré, Analyse non linéaire 5 (1988), 425-464. MR 90c:35084
  • [14] N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967) 473-484. MR 35:7121

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Additional Information

Shinji Adachi
Affiliation: Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan
Email: kazunaga@mn.waseda.ac.jp

Kazunaga Tanaka
Affiliation: Department of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan

DOI: https://doi.org/10.1090/S0002-9939-99-05180-1
Received by editor(s): May 5, 1998
Received by editor(s) in revised form: August 26, 1998
Published electronically: November 1, 1999
Additional Notes: The second author was partially supported by the Sumitomo Foundation (Grant No. 960354) and Waseda University Grant for Special Research Projects 97A-140, 98A-122.
Communicated by: Christopher Sogge
Article copyright: © Copyright 2000 American Mathematical Society

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