Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Concentration solutions to singularly prescribed Gaussian and geodesic
  curvatures problem   We consider the following Liouville-type equation with exponential Neumann
boundary condition: $$ -\Delta\tilde u = \varepsilon^2 K(x) e^{2\tilde u},
\quad x\in D, \qquad \frac{\partial \tilde u}{\partial n} + 1 = \varepsilon
\kappa(x) e^{\tilde u}, \quad x\in\partial D, $$ where $D\subset \mathbb R^2$
is the unit disc, $\varepsilon^2 K(x)$ and $\varepsilon \kappa(x)$ stand for
the prescribed Gaussian curvature and the prescribed geodesic curvature of the
boundary, respectively. We prove the existence of concentration solutions if
$\kappa(x) + \sqrt{K(x)+\kappa(x)^2}$ ($x\in\partial D$) has a strictly local
extremum point, which is a total new result for exponential Neumann boundary
problem.

15
The categories of this paper are: nucl-ex
3
The categories of this paper are: physics
11
The categories of this paper are: hep-lat
16
The categories of this paper are: stat
17
The categories of this paper are: q-fin
13
The categories of this paper are: quant-ph
14
The categories of this paper are: math-ph
24
The categories of this paper are: dg-ga
30
The categories of this paper are: q-alg
21
The categories of this paper are: adap-org
31
The categories of this paper are: ao-sci
32
The categories of this paper are: atom-ph
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th
12
The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio