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} (-\Delta)^{s}u = \lambda u^{-q} + u^{2^*_s-1}, \quad u>0 \quad \text{in }\Omega, \mathcal A(u) = 0 \quad \text{on}~ \partial\Omega = \sum_{D} \cup \sum_{\mathcal{N}}, \end{cases} \tag{$P_\lambda$} \end{equation*} where $\Omega \subset \mathbb{R}^N$ is a bounded domain with smooth boundary $\partial{\Omega}$, $1/2<s<1$, $\lambda >0$ is a real parameter, $0 < q < 1$, $N>2s$, $2^*_s=2N/(N-2s)$ and $$\mathcal{A}(u)= u \mathcal{X}_{\sum_{D}} + {\partial_{\nu}u}\mathcal{X}_{ \sum_{\mathcal{N}}}, \quad{\partial_{\nu}=\frac{\partial }{\partial{\nu}}}.$$ Here $\sum_{D}$, $\sum_{\mathcal{N}}$ are smooth $(N-1)$ dimensional submanifolds of $\partial \Omega$ such that $\sum_{D} \cup \sum_{\mathcal{N}}= \partial\Omega$, $\sum_{D} \cap \sum_{\mathcal{N}}= \emptyset$ and $\sum_{D} \cap \overline{\sum_{\mathcal{N}}} = \tau'$ is a smooth $(N-2)$ dimensional submanifold of $\partial{\Omega}$. 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.