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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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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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: http://dx.doi.org/10.1090/S0002-9939-99-05180-1
PII: S 0002-9939(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