Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A local pointwise inequality for a biharmonic equation with negative
  exponents   In this paper, we are inspired by Ng\^{o}, Nguyen and Phan's [15] study of
the pointwise inequality for positive $C^{4}$-solutions of biharmonic equations
with negative exponent by using the growth condition of solutions. They propose
an open question of whether the growth condition is necessary to obtain the
pointwise inequality. We give a positive answer to this open question. We
establish the following local pointwise inequality $$-\frac{\Delta
u}{u}+\alpha\frac{|\nabla u|^{2}}{u^{2}}+\beta
u^{-\frac{q+1}{2}}\leq\frac{C}{R^{2}}$$ for positive $C^{4}$-solutions of the
biharmonic equations with negative exponent $$-\Delta^{2}u=u^{-q} \ in \
B_{R}$$ where $B_{R}$ denotes the ball centered at $x_{0}$ with radius $R$,
$n\geq3$, $q>1$, and some constants $\alpha\geq0$, $\beta>0$, $C>0$.

23
The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
6
The categories of this paper are: astro-ph
31
The categories of this paper are: ao-sci
0
The categories of this paper are: hep-ph
26
The categories of this paper are: bayes-an
17
The categories of this paper are: q-fin
15
The categories of this paper are: nucl-ex
3
The categories of this paper are: physics
10
The categories of this paper are: q-bio
33
The categories of this paper are: plasm-ph
37
The categories of this paper are: supr-con
24
The categories of this paper are: dg-ga
35
The categories of this paper are: mtrl-th
20
The categories of this paper are: acc-phys
8
The categories of this paper are: hep-ex