Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A complete description of the asymptotic behavior at infinity of
  positive radial solutions to $\Delta^2 u = u^\alpha$ in $\mathbf R^n$   We consider the biharmonic equation $\Delta^2 u = u^\alpha$ in $\mathbf R^n$
with $n \geqslant 1$. It was proved that this equation has a positive classical
solution if, and only if, either $\alpha \leqslant 1$ with $n \geqslant 1$ or
$\alpha\geqslant (n+4)/(n-4)$ with $n \geqslant 5$. The asymptotic behavior at
infinity of all positive radial solutions was known in the case
$\alpha\geqslant (n+4)/(n-4)$ and $n \geqslant 5$. In this paper, we classify
the asymptotic behavior at infinity of all positive radial solutions in the
remaining case $\alpha\leqslant 1$ with $n \geqslant 1$; hence obtaining a
complete picture of the asymptotic behavior at infinity of positive radial
solutions. Since the underlying equation is of higher order, we propose a new
approach which relies on a representation formula and asymptotic analysis
arguments. We believe that the approach introduced here can be conveniently
applied to study other problems with higher order operators.

15
The categories of this paper are: nucl-ex
13
The categories of this paper are: quant-ph
3
The categories of this paper are: physics
18
The categories of this paper are: econ
16
The categories of this paper are: stat
28
The categories of this paper are: alg-geom
29
The categories of this paper are: funct-an
24
The categories of this paper are: dg-ga
17
The categories of this paper are: q-fin
33
The categories of this paper are: plasm-ph
23
The categories of this paper are: solv-int
4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th
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The categories of this paper are: eess
32
The categories of this paper are: atom-ph