  In a recent series of important works
\cite{wei-zhang-1,wei-zhang-2,wei-zhang-3}, Wei-Zhang 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.
