Multiplicity of solutions of Kirchhoff-type fractional Laplacian problems with critical and singular nonlinearities

In this ariticle, the following Kirchhoff-type fractional Laplacian problem with singular and critical nonlinearities is studied: \begin{equation*} \begin{cases} \left(a+b\|u\|^{2\mu-2}\right)\left(-\Delta\right)^su=\lambda l(x)u^{2_{s}^{*}-1}+h(x)u^{-\gamma},&\mbox{in}~\Omega,\\ u>0,&\mbox{in}~\Omega,\\ u=0,&\mbox{in}~\mathbb{R}^{N} \backslash\Omega, \end{cases} \end{equation*} where $s\in(0,1)$, $N>2s$, $(-\Delta)^{s}$ is the fractional Laplace operator, $2_{s}^{*}=2N/(N-2s)$ is the critical Sobolev exponent, $\Omega\subset\mathbb{R}^{N}$ is a smooth bounded domain, $l\in L^{\infty}(\Omega)$ is a non-negative function and $\max\left\{l(x),0\right\}\not\equiv0$, $h\in L^{\frac{2_{s}^{*}}{2_{s}^{*}+\gamma-1}}(\Omega)$ is positive almost everywhere in $\Omega$, $\gamma\in(0,1)$, $a>0,b>0$, $\mu\in\left[1,2_{s}^{*}/2\right)$ and parameter $\lambda$ is a positive constant. Here we utilize a special method to recover the lack of compactness due to the appearance of the critical exponent. By imposing appropriate constraint on $\lambda$, we obtain two positive solutions to the above problem based on the Ekeland variational principle and Nehari manifold technique