An improved Hardy-Trudinger-Moser inequality (1501.03678v2)

Abstract: Let $\mathbb{B}$ be the unit disc in $\mathbb{R}^2$, $\mathscr{H}$ be the completion of $C_0^\infty(\mathbb{B})$ under the norm $$|u|{\mathscr{H}}=\left(\int\mathbb{B}|\nabla u|^2dx-\int_\mathbb{B}\frac{u^2}{(1-|x|^2)^2}dx\right)^{1/2},\quad\forall u\in C_0^\infty(\mathbb{B}).$$ Denote $\lambda_1(\mathbb{B})=\inf_{u\in \mathscr{H},\,|u|2=1}|u|{\mathscr{H}}^2$, where $|\cdot|2$ stands for the $L^2(\mathbb{B})$-norm. Using blow-up analysis, we prove that for any $\alpha$, $0\leq \alpha<\lambda_1(\mathbb{B})$, $$\sup{u\in\mathscr{H},\,|u|{\mathscr{H}}^2-\alpha|u|_2^2\leq 1}\int\mathbb{B} e^{4\pi u^2}dx<+\infty,$$ and that the above supremum can be attained by some function $u\in \mathscr{H}$ with $|u|_{\mathscr{H}}^2-\alpha|u|_2^2= 1$. This improves an earlier result of G. Wang and D. Ye [28].