$-\mathcal L_K$ with homogeneous Dirichlet boundary data, there exists a left neighborhood $\mathcal O_k$ of $\lambda_k$ such that problem  admits at least three non-trivial weak solutions for all $\lambda\in \mathcal O_k$.

In the non-local framework, the simplest example we can deal with is given by the fractional Laplacian, according to the following result:

\[thlapfrac\] Let $s\in (0,1)$, $n>2s$ and $\Omega$ be an open bounded subset of ${\mathbb R}^n$ with continuous boundary. If $f:\Omega\times {\mathbb R}\to {\mathbb R}$ is a function verifying –, then the problem $$\label{problema}
\left\{
\begin{array}{ll}
(-\Delta)^s u-\lambda u=f(x,u) & {\mbox{ in }} \Omega\\
u=0 & {\mbox{ in }} {\mathbb R}^n\setminus \Omega
\end{array} \right.$$ admits at least three non-trivial weak solutions belonging to $H^s({\mathbb R}^n)$ in a suitable left neighborhood of any eigenvalue of $(-\Delta)^s$ with homogeneous Dirichlet boundary data.

When $s=1$, problem  reduces to a standard semilinear Laplace equation: in this sense Theorem \[thlapfrac\] may be seen as the fractional version of the result in [@mugnaiNODEA Theorem 1].

We prove Theorem \[thmain\] employing variational and topological methods. Precisely, we apply a $\nabla$-theorem due to Marino and Saccon in [@marsac:sns], see Theorem \[thmarinosaccon\] below. The main difficulties in applying such a theorem are obviously related to the nonlocal 