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} (-\Delta)^su=\lambda u+|u|^{2_s^*-2}u &\text{in}\ \Omega,\\ \mkern+38.5mu u=0& \text{on}\ \Sigma_{\mathcal{D}},\\ \mkern+24mu \displaystyle \frac{\partial u}{\partial \nu}=0 &\text{on}\ \Sigma_{\mathcal{N}}, \end{array} \right. \end{equation} where $(-\Delta)^s$, $s\in (1/2,1)$, is the spectral fractional Laplacian operator, $\Omega\subset\mathbb{R}^N$, $N>2s$, is a smooth bounded domain, $2_s^*=\frac{2N}{N-2s}$ denotes the critical fractional Sobolev exponent, $\lambda>0$ is a real parameter, $\nu$ is the outwards normal to $\partial\Omega$, $\Sigma_{\mathcal{D}}$, $\Sigma_{\mathcal{N}}$ are smooth $(N-1)$--dimensional submanifolds of $\partial\Omega$ such that $\Sigma_{\mathcal{D}}\cup\Sigma_{\mathcal{N}}=\partial\Omega$, $\Sigma_{\mathcal{D}}\cap\Sigma_{\mathcal{N}}=\emptyset$ and $\Sigma_{\mathcal{D}}\cap\overline{\Sigma}_{\mathcal{N}}=\Gamma$ is a smooth $(N-2)$--dimensional submanifold of $\partial\Omega$. By employing a $\nabla$-theorem we prove the existence of multiple solutions when the parameter $\lambda$ is in a left neighborhood of a given eigenvalue of $(-\Delta)^s$.