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

Source abstract

We study Trudinger type inequalities in R N {\mathbf {R}}^{N} and their best exponents α N \alpha _{N} . We show for α ∈ ( 0 , α N ) \alpha \in (0,\alpha _{N}) , α N = N ω N − 1 1 / ( N − 1 ) \alpha _{N}=N\omega _{N-1}^{1/(N-1)} ( ω N − 1 \omega _{N-1} is the surface area of the unit sphere in R N {\mathbf {R}}^{N} ), there exists a constant C α > 0 C_{\alpha }>0 such that ( ∗ ) ∫ R N Φ N ( α ( | u ( x ) | ‖ ∇ u ‖ L N ( R N ) ) N N − 1 ) d x ≤ C α ‖ u ‖ L N ( R N ) N ‖ ∇ u ‖ L N ( R N ) N $$\begin{equation*} \tag {$*$} \int _{\mathbf {R} ^{N}} \Phi _{N}\left (\alpha \left ( \frac {\left |u(x)\right | }{\|\nabla u\| _{L^{N}(\mathbf {R} ^{N})}} \right )^{\frac {N}{N-1}}\right )\, dx \leq C_{\alpha } \frac {\|u\|_{L^{N}(\mathbf {R} ^{N})} ^{N}}{\|\nabla u\|_{L^{N}(\mathbf {R} ^{N})}^{N}} \end{equation*}$$ for all u ∈ W 1 , N ( R N ) ∖ { 0 } u \in W^{1,N} (\mathbf {R} ^{N})\setminus \{ 0\} . Here Φ N ( ξ ) \Phi _{N}(\xi ) is defined by Φ N ( ξ ) = exp ⁡ ( ξ ) − ∑ j = 0 N − 2 1 j ! ξ j . $$\begin{equation*} \Phi _{N}(\xi ) = \exp (\xi ) - \sum _{j=0}^{N-2} {\frac {1}{j!}}\xi ^{j}. \end{equation*}$$ It is also shown that ( ∗ ) (*) with α ≥ α N \alpha \geq \alpha _{N} is false, which is different from the usual Trudinger’s inequalities in bounded domains.