  For Liouville equations with singular sources, the interpretation of the
equation and its impact are most significant if the singular sources are
quantized: the strength of each Dirac mass is a mutliple of $4\pi$. However the
study of bubbling solutions around a quantized singular source is particularly
challenging: near the singular source, the spherical Harnack inequality may not
hold and there are multiple local maximums of bubbling solutions all swarming
to the singular source. In this article we seek to provide a complete
understanding of the blowup picture in this core difficulty and we establish
two major types of results: First we prove that not only the first derivatives
of coefficient functions tend to zero at the singular source, the second
derivatives also have a vanishing estimate. Second we derive pointwise
estimates for bubbling solutions to be approximated by global solutions, which
have crucial applications in a number of important projects. Since bubbling
solutions near a singular source are very commonly observed in geometry and
physics, it seems that the estimates in this article can also be applied to
many related equations and systems with various backgrounds.
