Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the existence of multiple solutions for fractional Brezis Nirenberg
  type equations   The present paper studies the non-local fractional analogue of the famous
paper of Brezis and Nirenberg in [4]. Namely, we focus on the following model,
$$\begin{align*}\left(\mathcal{P}\right) \begin{cases} \left(-\Delta\right)^s
u-\lambda u &= \alpha |u|^{p-2}u + \beta|u|^{2^*-2}u \quad\mbox{in}\quad
\Omega,\\ u&=0\quad\mbox{in}\quad\mathbb{R}^N\setminus\Omega, \end{cases}
\end{align*}$$ where $(-\Delta)^s$ is the fractional Laplace operator, $s \in
(0,1)$, with $N \geq 3s$, $2<p<2^*$, $\beta>0, \lambda, \alpha \in \mathbb{R}$
and establish the existence of nontrivial solutions and sign-changing solutions
for the problem $(\mathcal{P})$.

6
The categories of this paper are: astro-ph
21
The categories of this paper are: adap-org
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The categories of this paper are: solv-int
12
The categories of this paper are: nucl-th
3
The categories of this paper are: physics
33
The categories of this paper are: plasm-ph
13
The categories of this paper are: quant-ph
22
The categories of this paper are: chao-dyn
18
The categories of this paper are: econ
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The categories of this paper are: supr-con
11
The categories of this paper are: hep-lat
14
The categories of this paper are: math-ph
17
The categories of this paper are: q-fin
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The categories of this paper are: comp-gas
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The categories of this paper are: nucl-ex
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The categories of this paper are: eess