Subcritical nonlocal problems with mixed boundary conditions

In this paper, by variational and topological arguments based on linking and $\mathrm{\nabla }$-theorems, we prove the existence of multiple solutions for the following nonlocal problem with mixed Dirichlet-Neumann boundary data, $$\{\begin{array}{lcl}(-\mathrm{\Delta }{)}^{s}u=\lambda u+f(x,u)& & \text{in }\mathrm{\Omega },\\ u=0& & \text{on }{\mathrm{\Sigma }}_{\mathcal{D}},\\ {\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }\nu }=0}& & \text{on }{\mathrm{\Sigma }}_{\mathcal{N}},\end{array}$$ 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, $\lambda >0$ is a real parameter, $\nu$ is the outward 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 $\mathrm{\partial }\mathrm{\Omega }$.