Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A note on the nonexistence of positive supersolutions to elliptic
  equations with gradient terms   We prove that if the elliptic problem $-\Delta u+b(x)|\nabla u|=c(x)u$ with
$c\ge0$ has a positive supersolution in a domain $\Omega$ of $ \IR^{N\ge 3}$,
then $c,b$ must satisfy the inequality \[\sqrt{ \int_\Omega c\phi^2}\le \sqrt{
\int_\Omega | \nabla\phi|^2}+\sqrt{ \int_\Omega \frac{b^2}{4}\phi^2},~~~\phi
\in C_c^\infty(\Omega).\] As an application, we obtain Liouville type theorems
for positive supersolutions in exterior domains when $c(x)-\frac{b^2(x)}{4}>0$
for large $|x|$, but unlike the known results we allow the case
$\liminf_{|x|\rightarrow\infty}c(x)-\frac{b^2(x)}{4}=0$. Also the weights $b$
and $c$ are allowed to be unbounded. In particular, among other things, we show
that if $\tau:=\limsup_{|x| \rightarrow\infty}|xb(x)|<\infty$ then this problem
does not admit any positive supersolution if
  \[\liminf_{|x| \rightarrow\infty}|x|^2c(x)> \frac{(N-2+\tau)^2}{4},\]
  and, when $\tau=\infty, $ we have the same if \[\limsup_{R\rightarrow\infty}
R\Big(\frac{ \inf_{R<|x|<2 R} (c(x)-\frac{b(x)^2}{4})}{\sup_{\frac{R}{2}<|x|<4
R}|b(x)|}\Big)=\infty.\]

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The categories of this paper are: physics
1
The categories of this paper are: cs
29
The categories of this paper are: funct-an
12
The categories of this paper are: nucl-th
19
The categories of this paper are: eess
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The categories of this paper are: astro-ph
33
The categories of this paper are: plasm-ph
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The categories of this paper are: cmp-lg
10
The categories of this paper are: q-bio
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The categories of this paper are: econ
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The categories of this paper are: nlin
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The categories of this paper are: gr-qc
34
The categories of this paper are: chem-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: acc-phys
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The categories of this paper are: adap-org