The Compactness of Extremals for a Singular Hardy-Trudinger-Moser Inequality

Abstract

Motivated by a recent work of Wang-Yang [19], we study the compactness of extremals $\left\{u_{\beta}\right\}$ for singular Hardy-Trudinger-Moser inequalities due to Hou [24]. In particular, by the method of blow-up analysis, we conclude that, up to a subsequence, $u_{\beta}$ converges to an extremal in some sense as β tends to zero.

Keywords

Compactness / Hardy-Trudinger-Moser inequality / blow-up analysis