Adams’ inequality with logarithmic weights in $\mathbb {R}^{4}$

Abstract:

Trudinger-Moser inequality with logarithmic weight was first established by Calanchi and Ruf [J. Differential Equations 258 (2015), pp. 1967–1989]. The aim of this paper is to address the higher order version; more precisely, we show the following inequality \begin{equation*} \sup _{u \in W_{0,rad}^{2,2}(B,\omega ),{{\left \| {\Delta u} \right \|}_\omega } \le 1} \int _B {\exp \left ( {\alpha {{\left | u \right |}^{\frac {2}{{1 - \beta }}}}} \right )} dx < + \infty \end{equation*} holds if and only if \[ \alpha \le {\alpha _\beta } = 4{\left [ {8{\pi ^2}\left ( {1 - \beta } \right )} \right ]^{\frac {1}{{1 - \beta }}}}, \] where $B$ denotes the unit ball in $\mathbb {R}^{4}$, $\beta \in \left ( {0,1} \right )$, $\omega \left ( x \right ) = {\left ( {\log \frac {1}{{\left | x \right |}}} \right )^\beta }$ or ${\left ( {\log \frac {e}{{\left | x \right |}}} \right )^\beta }$, and $W_{0,rad}^{2,2}(B,\omega )$ is the weighted Sobolev spaces. Our proof is based on a suitable change of variable that allows us to represent the laplacian of $u$ in terms of the second derivatives with respect to the new variable; this method was first used by Tarsi [Potential Anal. 37 (2012), pp. 353–385].