Simple blow-up solutions of singular Liouville equations

Abstract

In a recent series of important works Wei-Zhang [Adv. Math. 380 (2021), Paper No. 107606, 45; Proc. Lond. Math. Soc. (3) 124 (2022), pp. 106–131; Laplacian vanishing theorem for quantized singular Liouville equation, Preprint, arXiv:2202.10825, 2022] proved several vanishing theorems for non-simple blow-up solutions of singular Liouville equations. It is well known that a non-simple blow-up situation happens when the spherical Harnack inequality is violated near a quantized singular source. In this article, we further strengthen the conclusions of Wei-Zhang by proving that if the spherical Harnack inequality does hold, there exist blow-up solutions with non-vanishing coefficient functions.