Nonlocal critical problems with mixed boundary conditions and nearly resonant perturbations

We consider the following nonlocal critical problem with mixed Dirichlet-Neumann boundary conditions, \begin{equation} \left{ \begin{array}{ll} (-Δ)^su=λu+|u|^{2_s^*-2}u &$\text{in}$\ Ω,\ \mkern+38.5mu u=0& $\text{on}$\ Σ_{$\mathcal{D}$},\ \mkern+24mu \displaystyle \frac{\partial u}{\partial ν}=0 &$\text{on}$\ Σ_{$\mathcal{N}$}, \end{array} \right. \end{equation} where $(-\mathrm{\Delta }{)}^s$, $s\in (1\mathrm{/}2,1)$, is the spectral fractional Laplacian operator, $\mathrm{\Omega }\subset {\mathbb{R}}^N$, $N>2s$, is a smooth bounded domain, $2_s^{\ast }=\frac{2N}{N-2s}$ denotes the critical fractional Sobolev exponent, $\lambda >0$ is a real parameter, $\nu$ is the outwards normal to $\mathrm{\partial }\mathrm{\Omega }$, ${\mathrm{\Sigma }}_{\mathcal{D}}$, ${\mathrm{\Sigma }}_{\mathcal{N}}$ are smooth $(N-1)$–dimensional submanifolds of $\mathrm{\partial }\mathrm{\Omega }$ such that ${\mathrm{\Sigma }}_{\mathcal{D}}\cup {\mathrm{\Sigma }}_{\mathcal{N}}=\mathrm{\partial }\mathrm{\Omega }$, ${\mathrm{\Sigma }}_{\mathcal{D}}\cap {\mathrm{\Sigma }}_{\mathcal{N}}=\mathrm{\varnothing }$ and ${\mathrm{\Sigma }}_{\mathcal{D}}\cap {\bar{\mathrm{\Sigma }}}_{\mathcal{N}}=\mathrm{\Gamma }$ is a smooth $(N-2)$–dimensional submanifold of $\mathrm{\partial }\mathrm{\Omega }$. By employing a $\mathrm{\nabla }$-theorem we prove the existence of multiple solutions when the parameter $\lambda$ is in a left neighborhood of a given eigenvalue of $(-\mathrm{\Delta }{)}^s$.