An improved Hardy-Trudinger-Moser inequality

Abstract

Let $\mathbb{B}$ be the unit disc in ${\mathbb{R}}^2$, $\mathcal{H}$ be the completion of $C_0^{\infty }(\mathbb{B})$ under the norm $$\Vert u{\Vert }_{\mathcal{H}}={\left({\int }_{\mathbb{B}}|\nabla u{|}^{2}dx-{\int }_{\mathbb{B}}\frac{u^2}{(1-|x{|}^2{)}^2}dx\right)}^{1/2},\forall u\in C_0^{\infty }(\mathbb{B}).$$ Denote ${\lambda }_1(\mathbb{B})={\inf }_{u\in \mathcal{H},\Vert u{\Vert }_2=1}\Vert u{\Vert }_{\mathcal{H}}^2$, where $\Vert \cdot {\Vert }_2$ stands for the $L^2(\mathbb{B})$-norm. Using blow-up analysis, we prove that for any $\alpha$, $0\le \alpha <{\lambda }_1(\mathbb{B})$, $$\underset{u\in \mathcal{H},\Vert u{\Vert }_{\mathcal{H}}^2-\alpha \Vert u{\Vert }_2^2\le 1}{\sup }{\int }_{\mathbb{B}}{e}^{4\pi u^2}dx<+\infty ,$$ and that the above supremum can be attained by some function $u\in \mathcal{H}$ with $\Vert u{\Vert }_{\mathcal{H}}^2-\alpha \Vert u{\Vert }_2^2=1$. This improves an earlier result of G. Wang and D. Ye [28].

18 pages