Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Energy asymptotics in the three-dimensional Brezis--Nirenberg problem   For a bounded open set $\Omega\subset\mathbb R^3$ we consider the
minimization problem $$ S(a+\epsilon V) = \inf_{0\not\equiv u\in H^1_0(\Omega)}
\frac{\int_\Omega (|\nabla u|^2+ (a+\epsilon V) |u|^2)\,dx}{(\int_\Omega
u^6\,dx)^{1/3}} $$ involving the critical Sobolev exponent. The function $a$ is
assumed to be critical in the sense of Hebey and Vaugon. Under certain
assumptions on $a$ and $V$ we compute the asymptotics of $S(a+\epsilon V)-S$ as
$\epsilon\to 0+$, where $S$ is the Sobolev constant. (Almost) minimizers
concentrate at a point in the zero set of the Robin function corresponding to
$a$ and we determine the location of the concentration point within that set.
We also show that our assumptions are almost necessary to have $S(a+\epsilon
V)<S$ for all sufficiently small $\epsilon>0$.

13
The categories of this paper are: quant-ph
4
The categories of this paper are: cond-mat
24
The categories of this paper are: dg-ga
18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph
32
The categories of this paper are: atom-ph
33
The categories of this paper are: plasm-ph
23
The categories of this paper are: solv-int
12
The categories of this paper are: nucl-th
34
The categories of this paper are: chem-ph
6
The categories of this paper are: astro-ph
21
The categories of this paper are: adap-org
36
The categories of this paper are: cmp-lg
25
The categories of this paper are: patt-sol
14
The categories of this paper are: math-ph
27
The categories of this paper are: comp-gas