Extremals of singular Hardy–Trudinger–Moser inequality with remainder terms on unit disc

Let \(B\subset {\mathbb {R}}^2\) be the unit disc, and \({\mathcal {H}}\) be the completion of \(C_0^\infty ({B})\) under the norm

$$\begin{aligned} \Vert u\Vert _{{\mathcal {H}}}=\Bigg (\int _{{B}}|\nabla u|^2 {\textrm{d}}x- \int _{{B}}\frac{u^2}{(1-|x|^2)^2}{\textrm{d}}x\Bigg )^{\frac{1}{2}}. \end{aligned}$$

We derive in this paper extremals of singular Hardy–Trudinger–Moser inequality with remainder terms on B using the method of blow-up analysis and rearrangement argument: suppose \(0<t<2,\) there exists a constant \(\delta _0>0\) such that for \(\gamma \le 4\pi (1-t/2)+\delta _0\) the supremum

$$\begin{aligned} \sup _{u\in {\mathcal {H}},\Vert u\Vert _{{\mathcal {H}}}\le 1}\int _{{B}}\frac{{\textrm{e}}^{4\pi (1-t/2)u^2}-\gamma u^2}{|x|^t} {\textrm{d}}x \end{aligned}$$

can be attained. This extends results of Wang and Ye (Adv Math 230:294–320, 2012) and Yin (Bull Iran Math Soc 49, 2023).