Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Critical growth fractional elliptic problem with singular nonlinearities   In this article, we study the following fractional Laplacian equation with
critical growth and singular nonlinearity $$\quad (-\Delta)^s u = \lambda a(x)
u^{-q} + u^{2^*_s-1}, \quad u>0 \; \text{in}\; \Omega,\quad u = 0 \;
\mbox{in}\; \mathbb{R}^n \setminus\Omega,$$ where $\Omega$ is a bounded domain
in $\mathbb{R}^n$ with smooth boundary $\partial \Omega$, $n > 2s,\; s \in
(0,1),\; \lambda >0,\; 0 < q \leq 1 $, $\theta \leq a(x) \in L^\infty(\Omega)$,
for some $\theta>0$ and $2^*_s=\frac{2n}{n-2s}$. We use variational methods to
show the existence and multiplicity of positive solutions of the above problem
with respect to the parameter $\lambda$.

18
The categories of this paper are: econ
29
The categories of this paper are: funct-an
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The categories of this paper are: patt-sol
3
The categories of this paper are: physics
5
The categories of this paper are: gr-qc
4
The categories of this paper are: cond-mat
13
The categories of this paper are: quant-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: supr-con
8
The categories of this paper are: hep-ex
12
The categories of this paper are: nucl-th
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The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
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The categories of this paper are: hep-lat
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The categories of this paper are: q-alg