Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Nonlinear elliptic equations on the upper half space   In this paper we shall classify all positive solutions of $ \Delta u =a u^p$
on the upper half space $ H =\Bbb{R}_+^n$ with nonlinear boundary condition $
{\partial u}/{\partial t}= - b u^q $ on $\partial H$ for both positive
parameters $a, \ b>0$. We will prove that for $p \ge {(n+2)}/{(n-2)}, 1\leq
q<{n}/{(n-2)}$ (and $n \ge 3$) all positive solutions are functions of last
variable; for $p= {(n+2)}/{(n-2)}, q= {n}/{(n-2)}$ (and $n \ge 3$) positive
solutions must be either some functions depending only on last variable, or
radially symmetric functions.

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The categories of this paper are: hep-ph
19
The categories of this paper are: eess
33
The categories of this paper are: plasm-ph
15
The categories of this paper are: nucl-ex
30
The categories of this paper are: q-alg
7
The categories of this paper are: hep-th
21
The categories of this paper are: adap-org
10
The categories of this paper are: q-bio
14
The categories of this paper are: math-ph
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The categories of this paper are: stat
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The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
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The categories of this paper are: q-fin
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The categories of this paper are: gr-qc
32
The categories of this paper are: atom-ph
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The categories of this paper are: dg-ga