A Remark on Hardy-Trudinger-Moser Inequality (Journal of Partial Differential Equations, 2020)
Let $\mathbb{B}$ be the unit disc in $\mathbb{R}^2$, $\mathscr{H}$ be the completion of $C_0^{∞}(\mathbb{B})$ under the norm$$||u||_{\mathscr{H}}=\Big(\int_{\mathbb{B}}| abla u|^2dx-\int_{\mathbb{B}}\frac{u^2}{(1-|x|^2)^2}dx\Big)^{\frac{1}{2}},\quad \forall u\in C_0^{∞}(\mathbb{B}).$$Using blow-up analysis, we prove that for any $γ\leqslant 4π$, the supremum$$\begin{align*}\sup_{u\in\mathscr{H},||u||_{1,h}\leqslant 1}\int_{\mathbb{B}}e^{γ u^2}dx\end{align*}$$can be attained by some function $u_0\in \mathscr{H}$ with $||u_0||_{1,h}=1$, where $h$ is a decreasingly nonnegative, radially symmetric function, and satisfies a coercive condition. Namely there exists a constant $δ>0$ satisfying$$||u||_{1,h}^2=\|u\|_{\mathscr{H}}^2-\int_{\mathbb{B}}hu^2{\rm d}x≥ δ\|u\|_{\mathscr{H}}^2,\quad\forall \, u\in\mathscr{H}.$$This extends earlier results of Wang-Ye [1] and Yang-Zhu [2].
Keywords: Hardy-Trudinger-Moser inequality; Trudinger-Moser inequality; blow-up analysis.