any $j\in {\mathbb N}$.

With the previous estimates, we are now ready to show that $$\label{ujboundednabla}
\mbox{the sequence}\,\, \{\|u_j\|_{X_0}\}_{j\in{\mathbb N}}\,\, \mbox{is bounded in}\,\, {\mathbb R}.$$ For this it is enough to use and , which yield for any $j\in {\mathbb N}$ $$\label{fujnabla}
\begin{aligned}
\int_\Omega |f(x, u_j(x))|^{q/(q-1)}\,dx & \leq \int_\Omega \left(a_1+a_2|u_j(x)|^{q-1}\right)^{q/(q-1)} \\
& \leq \tilde a_1 +\tilde a_2 \int_\Omega |u_j(x)|^q\,dx\\
& \leq \tilde a_3 + \tilde a_4 \int_\Omega F(x,u_j(x))\,dx\,,
\end{aligned}$$ for suitable positive constants $\tilde a_i$, $i=1,\ldots, 4$. By , and we get assertion .

In view of and , we can assume that there exists $u_\infty\in \mathbb H_{k-1}\oplus \mathbb P_{k+m-1}$ such that $$\label{convergenze0nabla}
\begin{aligned}
 & \int_{{\mathbb R}^n\times {\mathbb R}^n}\big(u_j(x)-u_j(y)\big)\big(\varphi(x)-\varphi(y)\big) K(x-y)\,dx\,dy \to \\
& \qquad \qquad
\int_{{\mathbb R}^n\times {\mathbb R}^n}\big(u_\infty(x)-u_\infty(y)\big)\big(\varphi(x)-\varphi(y)\big)
K(x-y)\,dx\,dy  \quad \mbox{for any}\,\, \varphi\in X_0\,,
\end{aligned}$$ while, by [@svmountain Lemma 8] and [@brezis Theorem IV.9], up to a subsequence, $$\label{convergenze0bisnabla}
\begin{aligned}
& u_j \to u_\infty \quad \mbox{in}\,\, L^q({\mathbb R}^n)\\
& u_j \to u_\infty \quad \mbox{a.e. in}\,\, {\mathbb R}^n
\end{aligned}$$ as $j\to +\infty$ and there exists $\ell\in L^q({\mathbb R}^n)$ such that $$\label{dominata20nabla}
|u_j(x)|\leq \ell(x) \quad \mbox{a.e. in}\,\, {\mathbb R}^n\,\quad \mbox{for any}\,\,j\in {\mathbb N}\,.$$

Moreover, by and we get that for any $\varepsilon>0$ there exists $C_\varepsilon$ such that $$\label{assurdo1}
\begin{aligned}
0<\frac{\bar \sigma}{\tilde 