2017/02/25 by Youngae Lee, Lee, Youngae, Chang‐Shou Lin +5 · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1702.07854
openalex publication_date 2017/02/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.