Trudinger type inequalities in <IMG ALIGN=BOTTOM HEIGHT=17 WIDTH=29> and their best exponents期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Trudinger type inequalities in and their best exponents

Authors:	Shinji Adachi Kazunaga Tanaka

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

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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