Abstract In dimension n isolated singularities—at a finite point or at infinity—for solutions of total mass to the -Liouville equation are logarithmic type. As consequence, we simplify classification argument in Esposito (Anal Non Linéaire 35(3):781–801, 2018) and establish quantization result for entire singular equation.

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 see...

We present new n = (1, 1) and n = (1, 0) supersymmetric generalization of the Liouville equation, which originate from a geometrical approach to describing the classical dynamics of Green–Schwarz superstrings in N = 2, D = 3 and N = 1, D = 3 target superspace. Considered are a zero curvature representation and Bäcklund transformations associated with the supersymmetric non–linear equations. PAC...