Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Existence and nonexistence of least energy solutions of the Neumann
  problem for a semilinear elliptic equation with critical Sobolev exponent and
  a critical lower-order perturbation   Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^{N}$, with $N\geq 5$,
$a>0$, $\alpha\geq 0$ and $2^*=\frac{2N}{N-2}$. We show that the the exponent
$q=\frac{2(N-1)}{N-2}$ plays a critical role regarding the existence of least
energy (or ground state) solutions of the Neumann problem $$
\left\{\begin{array}{ll} -\Delta u+au=u^{2^*-1}-\alpha u^{q-1}&\mbox{in}\
\Omega,\\ u>0&\mbox{in}\ \Omega,\\ \frac{\partial u}{\partial\nu}=0&\mbox{on}\
\partial\Omega. \end{array}\right. $$ Namely, we prove that when
$q=\frac{2(N-1)}{N-2}$ there exists an $\alpha_{0}>0$ such that the problem has
a least energy solution if $\alpha<\alpha_{0}$ and has no least energy solution
if $\alpha>\alpha_{0}$.

15
The categories of this paper are: nucl-ex
30
The categories of this paper are: q-alg
16
The categories of this paper are: stat
3
The categories of this paper are: physics
9
The categories of this paper are: nlin
23
The categories of this paper are: solv-int
33
The categories of this paper are: plasm-ph
14
The categories of this paper are: math-ph
12
The categories of this paper are: nucl-th
17
The categories of this paper are: q-fin
24
The categories of this paper are: dg-ga
32
The categories of this paper are: atom-ph
21
The categories of this paper are: adap-org
1
The categories of this paper are: cs
25
The categories of this paper are: patt-sol
26
The categories of this paper are: bayes-an