Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Multiple positive solutions for degenerate Kirchhoff equations with
  singular and Choquard nonlinearity   In this paper we study the existence, multiplicity and regularity of positive
weak solutions for the following Kirchhoff-Choquard problem:
  \begin{equation*}
  \begin{array}{cc}
  \displaystyle M\left( \iint\limits_{\mathbb{R}^{2N}}
\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\,dxdy\right) (-\Delta)^s u =
\frac{\lambda}{u^\gamma} + \left( \int\limits_{\Omega} \frac{|u(y)|^{2^{*}_{\mu
,s}}}{|x-y|^ \mu}\, dy\right) |u|^{2^{*}_{\mu ,s}-2}u \;\text{in} \; \Omega,
  %\quad \quad u > 0\quad \text{in} \; \Omega,
  \quad \quad u = 0\quad \text{in} \; \mathbb{R}^{N}\backslash\Omega,
  \end{array}
  \end{equation*} where $\Omega$ is open bounded domain of $\mathbb{R}^{N}$
with $C^2$ boundary, $N > 2s$ and $s \in (0,1)$. $M$ models Kirchhoff-type
coefficient in particular, the degenerate case where Kirchhoff coefficient M is
zero at zero. $(-\Delta)^s$ is fractional Laplace operator, $\lambda > 0$ is a
real parameter, $\gamma \in (0,1)$ and $2^{*}_{\mu ,s} = \frac{2N-\mu}{N-2s}$
is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality.
We prove that each positive weak solution is bounded and satisfy H\"older
regularity of order $s$. Furthermore, using the variational methods and
truncation arguments we prove the existence of two positive solutions.

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The categories of this paper are: mtrl-th
29
The categories of this paper are: funct-an
16
The categories of this paper are: stat
23
The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
12
The categories of this paper are: nucl-th
26
The categories of this paper are: bayes-an
11
The categories of this paper are: hep-lat
24
The categories of this paper are: dg-ga
6
The categories of this paper are: astro-ph
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The categories of this paper are: ao-sci
8
The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat
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The categories of this paper are: chao-dyn
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The categories of this paper are: nlin
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The categories of this paper are: plasm-ph