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

Sharp estimates for solutions of mean field equation with collapsing singularity