Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Free boundary problems arising in the theory of maximal solutions of
  equations with exponential nonlinearities   We consider equations of the form $\Delta u +\lambda^2 V(x)e^{\,u}=\rho$ in
various two dimensional settings. We assume that $V>0$ is a given function,
$\lambda>0$ is a small parameter and $\rho=\mathcal O(1)$ or $\rho\to +\infty$
as $\lambda\to 0$. In a recent paper we prove the existence of the maximal
solutions for a particular choice $V\equiv 1$, $\rho=0$ when the problem is
posed in doubly connected domains under Dirichlet boundary conditions. We
related the maximal solutions with a novel free boundary problem. The purpose
of this note is to derive the corresponding free boundary problems in other
settings. Solvability of such problems is, viewed formally, the necessary
condition for the existence of the maximal solution.

36
The categories of this paper are: cmp-lg
3
The categories of this paper are: physics
7
The categories of this paper are: hep-th
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
28
The categories of this paper are: alg-geom
13
The categories of this paper are: quant-ph
26
The categories of this paper are: bayes-an
21
The categories of this paper are: adap-org
32
The categories of this paper are: atom-ph
25
The categories of this paper are: patt-sol
37
The categories of this paper are: supr-con
35
The categories of this paper are: mtrl-th
23
The categories of this paper are: solv-int
19
The categories of this paper are: eess
4
The categories of this paper are: cond-mat