Nonlocal critical exponent singular problems under mixed Dirichlet-Neumann boundary conditions

In this paper, we study the following singular problem, under mixed Dirichlet-Neumann boundary conditions, and involving the fractional Laplacian \begin{equation*} \label{1} \begin{cases} (-Δ)^{s}u = λu^{-q} + u^{2^s-1}, \quad u>0 \quad $\text{in }$Ω, \mathcal A(u) = 0 \quad $\text{on}$~ \partialΩ= \sum{D} \cup \sum_{$\mathcal{N}$}, \end{cases} \tag{$P_{\lambda }$} \end{equation} where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with smooth boundary $\mathrm{\partial }\mathrm{\Omega }$, $1\mathrm{/}2<s<1$, $\lambda >0$ is a real parameter, $0<q<1$, $N>2s$, $2_s^{\ast }=2N\mathrm{/}(N-2s)$ and $$\mathcal{A}(u)=u{\mathcal{X}}_{\sum _D}+{\mathrm{\partial }}_{\nu }u{\mathcal{X}}_{\sum _{\mathcal{N}}},{\mathrm{\partial }}_{\nu }=\frac{\mathrm{\partial }}{\mathrm{\partial }\nu }\mathrm{.}$$ Here ${\sum }_D$, ${\sum }_{\mathcal{N}}$ are smooth $(N-1)$ dimensional submanifolds of $\mathrm{\partial }\mathrm{\Omega }$ such that ${\sum }_D\cup {\sum }_{\mathcal{N}}=\mathrm{\partial }\mathrm{\Omega }$, ${\sum }_D\cap {\sum }_{\mathcal{N}}=\mathrm{\varnothing }$ and ${\sum }_D\cap \overline{{\sum }_{\mathcal{N}}}={\tau }^{\mathrm{\prime }}$ is a smooth $(N-2)$ dimensional submanifold of $\mathrm{\partial }\mathrm{\Omega }$. Within a suitable range of $\lambda$, we establish existence of at least two opposite energy solutions for \eqref{1} using the standard Nehari manifold technique.