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.
Categories: math.AP