Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Uniqueness for bubbling solutions with collapsing singularities   The seminal work \cite{bm} by Brezis and Merle showed that the bubbling
solutions of the mean field equation have the property of mass concentration.
Recently, Lin and Tarantello in \cite{lt} found that the "bubbling implies mass
concentration" phenomena might not hold if there is a collapse of
singularities. Furthermore, a sharp estimate \cite{llty} for the bubbling
solutions has been obtained. In this paper, we prove that there exists at most
one sequence of bubbling solutions if the collapsing singularity occurs. The
main difficulty comes from that after re-scaling, the difference of two
solutions locally converges to an element in the kernel space of the linearized
operator. It is well-known that the kernel space is three dimensional. So the
main technical ingredient of the proof is to show that the limit after
re-scaling is orthogonal to the kernel space.

37
The categories of this paper are: supr-con
30
The categories of this paper are: q-alg
8
The categories of this paper are: hep-ex
31
The categories of this paper are: ao-sci
6
The categories of this paper are: astro-ph
14
The categories of this paper are: math-ph
3
The categories of this paper are: physics
24
The categories of this paper are: dg-ga
33
The categories of this paper are: plasm-ph
4
The categories of this paper are: cond-mat
9
The categories of this paper are: nlin
34
The categories of this paper are: chem-ph
21
The categories of this paper are: adap-org
23
The categories of this paper are: solv-int
18
The categories of this paper are: econ
12
The categories of this paper are: nucl-th