via $\nabla$-theorems'
---

[^1]

Introduction {#sec:introduzione}
============

Classical critical point theorems, like the Mountain Pass Theorem, the Linking Theorem or the Saddle Point Theorem (see [@ar; @puccirad; @rabinowitz; @struwe]), have been extensively used in order to construct non-trivial solutions for nonlocal equations of the type $$\left\{
\begin{array}{ll}
(-\Delta)^s u=f(x,u) & {\mbox{ in }} \Omega\\
u=0 & {\mbox{ in }} {\mathbb R}^n\setminus \Omega\,
\end{array} \right.$$ under different growth assumptions on $f$ (see, e.g., [@colorado; @barrioscoloradoservadei; @capella; @fiscella; @fsvBS; @molicaservadei; @sY; @sBNRES; @svmountain; @svlinking; @servadeivaldinociBN; @servadeivaldinociBNLOW; @servadeivaldinociCFP; @tan] and references therein, and [@cabretan] for a minimization procedure). Here, $s\in (0,1)$ is fixed and $(-\Delta)^s$ is the fractional Laplace operator, which (up to normalization factors) may be defined as $$\label{2}
-(-\Delta)^s u(x)=
\int_{{\mathbb R}^n}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\,dy\,,
\,\,\,\,\, x\in {\mathbb R}^n\,.$$

The aim of this paper is to focus on the existence of multiple solutions for this kind of problems, in the case when $f$ is a superlinear and subcritical nonlinearity. Precisely, we will study the following problem: $$\label{problemaK}
\left\{
\begin{array}{ll}
\mathcal L_K u+\lambda u+f(x,u)=0 & {\mbox{ in }} \Omega\\
u=0 & {\mbox{ in }} {\mathbb R}^n\setminus \Omega.
\end{array} \right.$$ Here $s\in (0,1)$ is fixed, $n>2s$, $\Omega\subset {\mathbb R}^n$ is an open bounded set with continuous 