  In this paper, by variational and topological arguments based on linking and
$\nabla$-theorems, we prove the existence of multiple solutions for the
following nonlocal problem with mixed Dirichlet-Neumann boundary data, $$
\left\{
  \begin{array}{lcl}
  (-\Delta)^su=\lambda u+f(x,u) & &\text{in } \Omega, \\[2pt]
  \mkern+39mu u=0& &\text{on } \Sigma_{\mathcal{D}}, \\[2pt]
  \mkern+26mu \displaystyle \frac{\partial u}{\partial \nu}=0& &\text{on }
\Sigma_{\mathcal{N}},
  \end{array}
  \right. $$ 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, $\lambda>0$ is a real parameter, $\nu$ is the outward 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 $\partial\Omega$.
