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

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

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

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