Estimates for simple bubbling solutions of the Liouville equation with quantized singularities

Abstract For the Liouville equation with singular sources, the most subtle and challenging situation arises when the sources are quantized, namely, when each Dirac mass has a strength equal to a multiple of . In this regime, two types of blow‐up may occur: simple and nonsimple. In this paper, we concentrate on the simple blow‐up case and give complete and optimal estimates. We begin by establishing sharp estimates for a localized problem in a ball and then apply these estimates to the Dirichlet problem with multiple quantized sources. Our results show that the conditions used by del Pino, Esposito, and Musso to construct simple blow‐up solutions are not only sufficient but also necessary. We further obtain a sharp estimate on the total mass.