Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: p-th Kazdan-Warner equation on graph   Let $G=(V,E)$ be a connected finite graph and $C(V)$ be the set of functions
defined on $V$. Let $\Delta_p$ be the discrete $p$-Laplacian on $G$ with $p>1$
and $L=\Delta_p-k$, where $k\in C(V)$ is positive everywhere. Consider the
operator $L:C(V)\rightarrow C(V)$. We prove that $-L$ is one to one, onto and
preserves order. So it implies that there exists a unique solution to the
equation $Lu=f$ for any given $f\in C(V)$. We also prove that the equation
$\Delta_pu=\overline{f}-f$ has a solution which is unique up to a constant,
where $\overline{f}$ is the average of $f$. With the help of these results, we
finally give various conditions such that the $p$-th Kazdan-Warner equation
$\Delta_pu=c-he^u$ has a solution on $V$ for given $h\in C(V)$ and $c\in
\mathds{R}$. Thus we generalize Grigor'yan, Lin and Yang's work \cite{GLY} for
$p=2$ to any $p>1$.

32
The categories of this paper are: atom-ph
5
The categories of this paper are: gr-qc
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
10
The categories of this paper are: q-bio
21
The categories of this paper are: adap-org
7
The categories of this paper are: hep-th
30
The categories of this paper are: q-alg
4
The categories of this paper are: cond-mat
25
The categories of this paper are: patt-sol
28
The categories of this paper are: alg-geom
16
The categories of this paper are: stat
37
The categories of this paper are: supr-con
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
3
The categories of this paper are: physics