boundary, and the non-local operator $\mathcal L_K$ is defined as $$\label{lk}
\mathcal L_Ku(x)=
\int_{{\mathbb R}^n}\Big(u(x+y)+u(x-y)-2u(x)\Big)K(y)\,dy\,,
\,\,\,\,\, x\in {\mathbb R}^n,$$ and $K:{\mathbb R}^n\setminus\{0\}\rightarrow(0,+\infty)$ is a function with the properties that $$\label{kernel}
{\mbox{$m K\in L^1({\mathbb R}^n)$, where $m(x)=\min \{|x|^2, 1\}$\,;}}$$ $$\label{kernelfrac}
\mbox{there exists}\,\, \theta>0\,\,
\mbox{such that}\,\, K(x)\geq \theta |x|^{-(n+2s)}\,\,
\mbox{for any}\,\, x\in {\mathbb R}^n \setminus\{0\}\,.\\$$ A model for $K$ is given by $K(x)=|x|^{-(n+2s)}$. In this case $\mathcal L_K$ is the fractional Laplace operator $-(-\Delta)^s$.

In the recent papers [@fiscellamolicaservadeiCFS; @152; @molicaMRL; @servadeiKavian] the multiplicity of solutions in the nonlocal fractional setting has been addressed by means of classical critical point theorems in the spirit of the ones cited above.

In [@marsac:sns] (see also [@MMu; @marsac]) Marino and Saccon introduced new critical point theorems, the so-called *theorems of mixed type*, or *$\nabla$-theorems*, which allow to get multiplicity results for semilinear elliptic problems (see, for instance, [@mms; @mug4; @mugnaiNODEA; @Mug; @mugpag; @OL; @fwang; @fwang2; @wzz]).

We think that a natural question is whether or not these techniques may be adapted to the fractional Laplacian setting. One can define a fractional power of the Laplacian using its spectral decomposition: indeed, in [@mugpag] the same problem considered along this paper, but for this spectral fractional Laplacian, 