Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Positive solutions to multi-critical elliptic problems   In this paper, we investigate the existence of multiple solutions to the
following multi-critical elliptic problem \begin{equation}\label{eq:0.1}
\left\{\begin{aligned} -\Delta u & =\lambda |u|^{p-2}u
+\sum_{i=1}^k(|x|^{-(N-\alpha_i)}*|u|^{2^*_i})|u|^{2^*_i-2}u\quad {\rm in}\quad
\Omega,\\ &u\in H^1_0(\Omega)\\ \end{aligned}\right. \end{equation} in
connection with the topology of the bounded domain $\Omega\subset \mathbb{R}^N,
\,N\geq 4$, where $\lambda>0$, $2^*_i=\frac{N+\alpha_i}{N-2}$ with
$N-4<\alpha_i<N,\ \ i=1,2,\cdot\cdot\cdot, k$ are critical
Hardy-Littlewood-Sobolev exponents and $2<p<22^*_{min}$ with
$2^*_{min}=\min\{2^*_i, \ i=1,2,\cdot\cdot\cdot, k\}$. We show that there is
$\lambda^*>0$ such that if $0<\lambda<\lambda^*$ problem \eqref{eq:0.1}
possesses at least $cat_\Omega(\Omega)$ positive solutions. We also study the
existence and uniqueness of solutions for the limit problem of \eqref{eq:0.1}.

8
The categories of this paper are: hep-ex
19
The categories of this paper are: eess
21
The categories of this paper are: adap-org
25
The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat
23
The categories of this paper are: solv-int
35
The categories of this paper are: mtrl-th
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg
31
The categories of this paper are: ao-sci
26
The categories of this paper are: bayes-an
20
The categories of this paper are: acc-phys
37
The categories of this paper are: supr-con
18
The categories of this paper are: econ
32
The categories of this paper are: atom-ph
1
The categories of this paper are: cs