# Sharp estimates for solutions of mean field equation with collapsing singularity

## TL;DR

This paper investigates blow-up solutions of mean field equations with collapsing singularities, providing explicit examples and sharp estimates that reveal new non-concentration phenomena and the role of collapsing rates as blow-up parameters.

## Contribution

It introduces explicit examples of non-concentration in collapsing singularities and derives precise estimates, advancing understanding of blow-up behavior in mean field equations.

## Key findings

- Non-concentration phenomena can occur during singularity collapse.
- Collapsing rate can serve as a blow-up parameter.
- Sharp estimates around blow-up points are established.

## 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.