Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On singular equations with critical and supercritical exponents   We study the problem \begin{equation*} (I_{\epsilon}) \left\{\begin{aligned}
  -\Delta u- \frac{\mu u}{|x|^2}&=u^p -\epsilon u^q \quad\text{in }\quad
\Omega, \\ u&>0 \quad\text{in }\quad \Omega, \\ u &\in H^1_0(\Omega)\cap
L^{q+1}(\Omega),
  \end{aligned}
  \right. \end{equation*} where $q>p\geq 2^*-1$, $\epsilon>0$ is a parameter,
$\Omega\subseteq\mathbb{R}^N$ is a bounded domain with smooth boundary, $0\in
\Omega$, $N\geq 3$ and $0<\mu<\bar\mu:=\big(\frac{N-2}{2}\big)^2$. We prove at
$0$, any solution of $(I_{\epsilon})$ has the singularity of order $|x|^{-\nu}$
when $q<\frac{2+\nu}{\nu}$ and of the order $|x|^{-\frac{2}{q-1}}$, when
$q>\frac{2+\nu}{\nu}$, where $\nu=\sqrt{\bar\mu}-\sqrt{\bar\mu-\mu}$. Moreover,
we show that when $q=\frac{2+\nu}{\nu}$ and $u$ is radial, $u\sim
|x|^{-\nu}|\log|x||^{-\frac{\nu}{2}}$. This gives the complete classification
of singularity at $0$ in the supercritical case. We also obtain gradient
estimate. Using the transformation $v=|x|^{\nu}u$, we reduce the problem
$(I_{\epsilon})$ to $(J_{\epsilon})$ \begin{equation*} (J_{\epsilon})
\left\{\begin{aligned}
  -div(|x|^{-2\nu} \nabla v)&=|x|^{-(p+1)\nu} v^p -\epsilon |x|^{-(q+1)\nu} v^q
\quad\text{in }\quad \Omega, \\ v&>0 \quad\text{in }\quad \Omega, \\ v& \in
H^1_0(\Omega, |x|^{-2\nu} )\cap L^{q+1}(\Omega, |x|^{-(q+1)\nu} ),
  \end{aligned}
  \right. \end{equation*} and then formulating a variational problem for
$(J_{\epsilon})$, we establish the existence of a variational solution
$v_{\epsilon}$. Furthermore, we characterize the asymptotic behavior of
$v_{\epsilon}$ as $\epsilon\to 0$ by variational arguments and when $p=2^*-1$,
we show how the solution $v_{\epsilon}$ blows-up at $0$.
  This is the first paper where the results have been established with super
critical exponents for $\mu>0$.

13
The categories of this paper are: quant-ph
33
The categories of this paper are: plasm-ph
19
The categories of this paper are: eess
3
The categories of this paper are: physics
4
The categories of this paper are: cond-mat
5
The categories of this paper are: gr-qc
17
The categories of this paper are: q-fin
32
The categories of this paper are: atom-ph
8
The categories of this paper are: hep-ex
14
The categories of this paper are: math-ph
36
The categories of this paper are: cmp-lg
12
The categories of this paper are: nucl-th
26
The categories of this paper are: bayes-an
10
The categories of this paper are: q-bio
25
The categories of this paper are: patt-sol
37
The categories of this paper are: supr-con