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

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.