Title: Sharp estimates for solutions of mean field equation with collapsing singularity,
Abstract: The pioneering work of Brezis-Merle [7], Li-Shafrir [27], Li [26] and
Bartolucci-Tarantello [4] showed that any sequence of blow up solutions for
(singular) mean field equations of Liouville type must exhibit a "mass
concentration" property. A typical situation of blow-up occurs when we let the
singular (vortex) points involved in the equation (see (1.1) below) collapse
together. However in this case Lin-Tarantello in [30] pointed out that the
phenomenon: "bubbling implies mass concentration" might not occur and new
scenarios open for investigation. In this paper, we present two explicit
examples which illustrate (with mathematical rigor) how a "non-concentration"
situation does happen and its new features. Among other facts, we show that in
certain situations, the collapsing rate of the singularities can be used as
blow up parameter to describe the bubbling properties of the solution-sequence.
In this way we are able to establish accurate estimates around the blow-up
points which we hope to use towards a degree counting formula for the shadow
system (1.34) below.