Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Gaussian curvature equations in $\mathbb{R}^2$ with prescribed
  non-positive curvature   The purpose of this paper is to study the solutions of $$ \Delta u +K(x)
e^{2u}=0 \quad{\rm in}\;\; \mathbb{R}^2 $$ with $K\le 0$. We introduce the
following quantity: $$\alpha_p(K)=\sup\left\{\alpha \in \mathbb{R}:\,
\int_{\mathbb{R}^2} |K(x)|^p(1+|x|)^{2\alpha p+2(p-1)} dx<+\infty\right\},
\quad \forall\; p \ge 1.$$ Under the assumption $({\mathbb H}_1)$:
$\alpha_p(K)> -\infty$ for some $p>1$ and $\alpha_1(K) > 0$, we show that for
any $0 < \alpha < \alpha_1(K)$, there is a unique solution $u_\alpha$ with
$u_\alpha(x) = \alpha \ln |x|+ c_\alpha+o\big(|x|^{-\frac{2\beta}{1+2\beta}}
\big)$ at infinity and $\beta\in (0,\,\alpha_1(K)-\alpha)$. Furthermore, we
show an example $K_0 \leq 0$ such that $\alpha_p(K_0) = -\infty$ for any $p>1$
and $\alpha_1(K_0) > 0$, for which we study the asymptotic behavior of
solutions. In particular, we prove the existence of a solution $u_*$ such that
$u_* -\alpha_*\ln|x| = O(1)$ at infinity for some $\alpha_* > 0$, but who does
not converge to a constant at infinity. This example exhibits a new phenomenon
of solutions with logarithmic growth and non-uniform behavior at infinity.

12
The categories of this paper are: nucl-th
35
The categories of this paper are: mtrl-th
11
The categories of this paper are: hep-lat
1
The categories of this paper are: cs
14
The categories of this paper are: math-ph
8
The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph
16
The categories of this paper are: stat
7
The categories of this paper are: hep-th
6
The categories of this paper are: astro-ph
30
The categories of this paper are: q-alg
21
The categories of this paper are: adap-org
36
The categories of this paper are: cmp-lg
37
The categories of this paper are: supr-con
4
The categories of this paper are: cond-mat
10
The categories of this paper are: q-bio