there exists $\bar \sigma>0$, a sequence $\{\mu_j\}_{j\in {\mathbb N}}$ in ${\mathbb R}$ with $$\label{lambdaj}
\mu_j\in [\lambda_{k-1}+\bar \sigma, \lambda_{k+m}-\bar \sigma]$$ and a sequence $\{u_j\}_{j\in {\mathbb N}}$ such that $$\label{ujoplus}
u_j\in \mathbb H_{k-1}\oplus \mathbb P_{k+m-1}\setminus\{0\}\,,$$ $$\label{ujvincolato}
\langle\,\mathcal J'_{\mu_j}(u_j),\varphi\,\rangle =0\qquad \mbox{for any}\,\, \varphi\in \mathbb H_{k-1}\oplus \mathbb P_{k+m-1}\,,$$ for any $j\in {\mathbb N}$, and $$\label{ujvalorecritico}
\begin{aligned}
\mathcal J_{\mu_j}(u_j) & =\frac 1 2 \int_{{\mathbb R}^n\times {\mathbb R}^n}|u_j(x)-u_j(y)|^2 K(x-y)\,dx\,dy -\frac{\mu_j}{2} \int_\Omega |u_j(x)|^2\,dx\\
& \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad- \int_\Omega F(x, u_j(x))\,dx\to 0
\end{aligned}$$ as $j\to +\infty$.

Taking $\varphi=u_j$ in (this is possible thanks to ) and using , we get that for any $j\in {\mathbb N}$ $$\begin{aligned}
0 & =\int_{{\mathbb R}^n\times {\mathbb R}^n}|u_j(x)-u_j(y)|^2 K(x-y)\,dx\,dy -\mu_j \int_\Omega |u_j(x)|^2\,dx- \int_\Omega f(x, u_j(x))u_j(x)\,dx\\
& = 2\mathcal J_{\mu_j}(u_j)+\int_\Omega \Big(2F(x, u_j(x))-f(x, u_j(x))u_j(x)\Big)\,dx\\
& \leq 2\mathcal J_{\mu_j}(u_j)+(2-q)\int_\Omega F(x, u_j(x))\,dx\,.
\end{aligned}$$ Hence, by this inequality, the fact that $q>2$ and again , we deduce that $$0<(q-2)\int_\Omega F(x, u_j(x))\,dx\leq 2\mathcal J_{\mu_j}(u_j)\to 0$$ thanks to , so that we get $$\label{Fujlimite}
\int_\Omega F(x, u_j(x))\,dx\to 0$$ as $j\to +\infty$.

Now, since holds true, for any $j\in{\mathbb N}$ there exist $v_j\in \mathbb H_{k-1}$ and $w_j\in\mathbb P_{k+m-1}$ such that $$u_j=v_j+w_j\,.$$ Letting $\varphi=v_j-w_j$ in and taking into account the orthogonality properties of $v_j$ and $w_j$, we 