has been considered. As in [@mugpag], the purpose of this paper is to investigate the existence of multiple weak solutions for .

A weak solution of is a solution of the following problem: $$\label{problemaKweak}
\left\{\begin{array}{l}
{\displaystyle \int_{{\mathbb R}^n \times {\mathbb R}^n } (u(x)-u(y))(\varphi(x)-\varphi(y))K(x-y) dx\,dy-\lambda \int_\Omega u(x)\varphi(x)\,dx}\\
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
{\displaystyle = \int_\Omega f(x,u(x))\varphi(x)dx}\,\,\,\,\,\,\, \forall\,\, \varphi \in X_0\\
\\
u\in X_0
\end{array}\right.$$ (for this see [@sv Lemma 5.6] and [@servadeivaldinociBN footnote 3]). Here $X_0$ is defined as follows: $X$ is the linear space of Lebesgue measurable functions from ${\mathbb R}^n$ to ${\mathbb R}$ such that the restriction to $\Omega$ of any function $g$ in $X$ belongs to $L^2(\Omega)$ and $$\mbox{the map}\,\,\,
(x,y)\mapsto (g(x)-g(y))\sqrt{K(x-y)}\,\,\, \mbox{is in}\,\,\,
L^2\big({\mathbb R}^n \times {\mathbb R}^n \setminus ({\mathcal C}\Omega\times
{\mathcal C}\Omega), dxdy\big)\,,$$ where ${\mathcal C}\Omega:={\mathbb R}^n \setminus\Omega$. Moreover, $$X_0=\{g\in X : g=0\,\, \mbox{a.e. in}\,\,
{\mathbb R}^n\setminus
\Omega\}\,.$$

As we said here above, we suppose that equation  is superlinear and subcritical, that is its right-hand side $f:\Omega\times {\mathbb R}\to {\mathbb R}$ verifies the following conditions: $$\label{caratheodory}
f\,\, \mbox{is a Carath\'eodory function};$$ $$\label{crescita}
\begin{aligned}
&\mbox{there exist}\,\, a_1, a_2>0\,\,\mbox{and}\,\, q\in (2, 2^*), 2^*=2n/(n-2s)\,,\,\, \mbox{such that}\\
&\qquad \qquad |f(x,t)|\le a_1+a_2|t|^{q-1}\,\, \mbox{a.e.}\,\, x\in \Omega, t\in {\mathbb R}\,;
