Estimates for Liouville equation with quantized singularities

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