Sharp estimates for solutions of mean field equations with collapsing singularity

Abstract

The pioneering work of Brezis-Merle [7], Li-Shafrir [27], Li [26], and Bartolucci-Tarantello [3] 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 blowup 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 nonconcentration 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 toward a degree counting formula for the shadow system (1.34) below.