Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Blow-up phenomena for the Liouville equation with a singular source of
  integer multiplicity   We are concerned with the existence of blowing-up solutions to the following
boundary value problem $$-\Delta u= \la a(x) e^u-4\pi N \delta_0\;\hbox{ in }
\Omega,\quad u=0 \;\hbox{ on }\partial \Omega,$$ where $\Omega$ is a smooth and
bounded domain in $\R^2$ such that $0\in\Omega$, $a(x)$ is a positive smooth
function, $N$ is a positive integer and $\la>0$ is a small parameter. Here
$\delta_0$ defines the Dirac measure with pole at $0$. We find conditions on
the function $a$ and on the domain $\Omega$ under which there exists a solution
$u_\la$ blowing up at $0$ and satisfying $\la\into a(x)e^{u_\la} \to 8\pi(N+1)$
as $\la\to 0^+$.

3
The categories of this paper are: physics
19
The categories of this paper are: eess
26
The categories of this paper are: bayes-an
23
The categories of this paper are: solv-int
29
The categories of this paper are: funct-an
24
The categories of this paper are: dg-ga
30
The categories of this paper are: q-alg
12
The categories of this paper are: nucl-th
8
The categories of this paper are: hep-ex
36
The categories of this paper are: cmp-lg
18
The categories of this paper are: econ
1
The categories of this paper are: cs
27
The categories of this paper are: comp-gas
31
The categories of this paper are: ao-sci
20
The categories of this paper are: acc-phys
22
The categories of this paper are: chao-dyn