Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Exponential elliptic boundary value problems on a solid torus in the
  critical of supercritical case   In this paper we investigate the behavior and the existence of positive and
non-radially symmetric solutions to nonlinear exponential elliptic model
problems defined on a solid torus $\bar{T}$ of $\mathbb{R}^3$, when data are
invariant under the group $G=O(2)\times I \subset O(3)$. The model problems of
interest are stated below: ${ll} {\bf(P_1)} & \displaystyle
  \Delta\upsilon+\gamma=f(x)e^\upsilon, \upsilon>0\quad \mathrm{on} \quad T,
\quad\upsilon |_{_{\partial T}}=0.$ and ${ll}\bf{(P_2)} & \displaystyle
\Delta\upsilon+a+fe^\upsilon=0, \upsilon>0\quad \mathrm{on}\quad T, [1.3ex]
&\displaystyle \frac{\partial \upsilon}{\partial n}+b+ge^\upsilon=0\quad
\mathrm{on} \quad{\partial T}.$ We prove that exist solutions which are
$G-$invariant and these exhibit no radial symmetries. In order to solve the
above problems we need to find the best constants in the Sobolev inequalities
in the exceptional case.

29
The categories of this paper are: funct-an
37
The categories of this paper are: supr-con
19
The categories of this paper are: eess
21
The categories of this paper are: adap-org
18
The categories of this paper are: econ
24
The categories of this paper are: dg-ga
22
The categories of this paper are: chao-dyn
9
The categories of this paper are: nlin
25
The categories of this paper are: patt-sol
15
The categories of this paper are: nucl-ex
12
The categories of this paper are: nucl-th
27
The categories of this paper are: comp-gas
35
The categories of this paper are: mtrl-th
28
The categories of this paper are: alg-geom
23
The categories of this paper are: solv-int
3
The categories of this paper are: physics