|u(x)|^2\,dx-\int_\Omega F(x, u(x))dx\,.$$ Note that when functional $\mathcal J_\lambda$ is Fréchet differentiable at $u\in X_0$, we have that for any $\varphi\in X_0$ $$\begin{aligned}
\langle \mathcal J'_\lambda(u), \varphi\rangle & = \int_{{\mathbb R}^n\times {\mathbb R}^n} \big(u(x)-u(y)\big)\big(\varphi(x)-\varphi(y)\big)K(x-y)\,dx\,dy-\lambda \int_\Omega u(x)\varphi(x)\,dx\\
& \qquad \qquad \qquad \qquad \qquad \qquad\qquad \qquad \qquad \qquad  -\int_\Omega f(x, u(x))\varphi(x)\,dx\,,
\end{aligned}$$ where we have denoted by $\langle \cdot,\cdot\rangle$ the duality between $X_0'$ and $X_0$. Thus, critical points of $\mathcal J_\lambda$ are solutions to problem . In order to find multiplicity of critical points, we will use the $\nabla$-theorem in the form of Theorem \[thmarinosaccon\] (see Section \[sec:proofthmain\]).

Estimates on the nonlinearity
-----------------------------

Here, we recall some estimates on the nonlinear term and its primitive, which will be useful in the sequel. These estimates are quite standard and do not take into account the non-local features of the problem. For a proof we refer to [@svmountain Lemma 3 and Lemma 4].

By assumptions , and we deduce that $$\label{cond22}
\begin{aligned}
& \mbox{for any}\,\, \varepsilon>0\,\, \mbox{there exists}\,\, C_{\varepsilon}>0\,\, \mbox{such that}\\
& \quad |f(x,t)|\leq 2\varepsilon |t|+qC_{\varepsilon} |t|^{q-1}\,\, \mbox{a.e.}\,\, x\in \Omega, t\in {\mathbb R}\end{aligned}$$ and so, as a consequence, $$\label{cond22F}
|F(x,t)|\leq \varepsilon\,|t|^2+C_{\varepsilon}\, |t|^{q}\,,$$ where $F$ is defined as in . This implies that functional $\mathcal 