Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Existence and stability properties of entire solutions to the
  polyharmonic equation $(-\Delta)^m u=e^u$ for any $m\ge 1$   We study existence and stability properties of entire solutions of a
polyharmonic equation with an exponential nonlinearity. We study existence of
radial entire solutions and we provide some asymptotic estimates on their
behavior at infinity. As a first result on stability we prove that stable
solutions (not necessarily radial) in dimensions lower than the conformal one
never exist. On the other hand, we prove that radial entire solutions which are
stable outside a compact always exist both in high and low dimensions. In order
to prove stability of solutions outside a compact set we prove some new
Hardy-Rellich type inequalities in low dimensions.

20
The categories of this paper are: acc-phys
28
The categories of this paper are: alg-geom
34
The categories of this paper are: chem-ph
29
The categories of this paper are: funct-an
10
The categories of this paper are: q-bio
25
The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex
21
The categories of this paper are: adap-org
18
The categories of this paper are: econ
36
The categories of this paper are: cmp-lg
3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
0
The categories of this paper are: hep-ph
11
The categories of this paper are: hep-lat
6
The categories of this paper are: astro-ph
30
The categories of this paper are: q-alg