Adams’ inequality with logarithmic weights in ℝ

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 sup u ∈ W 0 , r a d 2 , 2 ( B , ω ) , ‖ Δ u ‖ ω ≤ 1 ∫ B exp ⁡ ( α | u | 2 1 − β ) d x > + ∞ \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 \[ α ≤ α β = 4 [ 8 π 2 ( 1 − β ) ] 1 1 − β , \alpha \le {\alpha _\beta } = 4{\left [ {8{\pi ^2}\left ( {1 - \beta } \right )} \right ]^{\frac {1}{{1 - \beta }}}, \] where B B denotes the unit ball in R 4 \mathbb {R}^{4} , β ∈ ( 0 , 1 ) \beta \in \left ( {0,1} \right ) , ω ( x ) = ( log ⁡ 1 | x | ) β \omega \left ( x \right ) = {\left ( {\log \frac {1}{{\left | x \right |}}} \right )^\beta } or ( log ⁡ e | x | ) β {\left ( {\log \frac {e}{{\left | x \right |}}} \right )^\beta } , and W 0 , r a d 2 , 2 ( B , ω ) 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 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].