have that for any $j\in {\mathbb N}$ $$\label{split}
\begin{aligned}
0 & =\langle\mathcal J'_{\mu_j}(u_j), v_j-w_j\rangle\\
& = \int_{{\mathbb R}^n\times {\mathbb R}^n}|v_j(x)-v_j(y)|^2 K(x-y)\,dx\,dy- \int_{{\mathbb R}^n\times {\mathbb R}^n}|w_j(x)-w_j(y)|^2 K(x-y)\,dx\,dy\\
& \qquad \qquad -\mu_j \int_\Omega |v_j(x)|^2\,dx + \mu_j \int_\Omega |w_j(x)|^2\,dx- \int_\Omega f(x, u_j(x))(v_j(x)-w_j(x))\,dx\,.
\end{aligned}$$ By and , gives that for any $j\in {\mathbb N}$ $$\label{carvar2}
\begin{aligned}
\int_\Omega f(x, u_j(x))(v_j(x)-w_j(x))\,dx & = \int_{{\mathbb R}^n\times {\mathbb R}^n}|v_j(x)-v_j(y)|^2 K(x-y)\,dx\,dy\\
& \qquad \qquad - \int_{{\mathbb R}^n\times {\mathbb R}^n}|w_j(x)-w_j(y)|^2 K(x-y)\,dx\,dy\\
& \qquad \qquad -\mu_j \int_\Omega |v_j(x)|^2\,dx + \mu_j \int_\Omega |w_j(x)|^2\,dx\\
& \leq \frac{\lambda_{k-1}-\mu_j}{\lambda_{k-1}}\|v_j\|_{X_0}^2+ \frac{\mu_j-\lambda_{k+m}}{\lambda_{k+m}}\|w_j\|_{X_0}^2\\
& \leq -\frac{\bar \sigma}{\lambda_{k-1}}\|v_j\|_{X_0}^2- \frac{\bar \sigma}{\lambda_{k+m}}\|w_j\|_{X_0}^2\\
&\leq -\frac{\bar \sigma}{\lambda_{k+m}}\|u_j\|_{X_0}^2\,,
\end{aligned}$$ thanks again to the properties of the projections $v_j$ and $w_j$ of $u_j$, respectively on $\mathbb H_{k-1}$ and on $\mathbb P_{k+m-1}$, and to .

On the other hand, by the Hölder inequality, and the fact that $X_0\hookrightarrow L^q(\Omega)$ compactly, there exists a suitable positive constant $\tilde\kappa$, independent of $j$, such that $$\begin{aligned}
\Big|\int_\Omega f(x, u_j(x))(v_j(x)-w_j(x))\,dx\Big| & \leq
\|f(\cdot, u_j(\cdot))\|_{L^{q/(q-1)}(\Omega)}\, \|v_j-w_j\|_{L^q(\Omega)}\\
& \leq  \tilde\kappa \|f(\cdot, u_j(\cdot))\|_{L^{q/(q-1)}(\Omega)}\, \|v_j-w_j\|_{X_0}\\
& = \tilde\kappa \|f(\cdot, u_j(\cdot))\|_{L^{q/(q-1)}(\Omega)}\, \|u_j\|_{X_0}\,,\\
\end{aligned}$$ so that, by this and , we deduce that $$\tilde\kappa \|f(\cdot, u_j(\cdot))\|_{L^{q/(q-1)}(\Omega)}\, \|u_j\|_{X_0}\geq \frac{\bar \sigma}{\lambda_{k+m}}\|u_j\|_{X_0}^2.$$ Being $u_j\not \equiv 0$ by assumption (see ), we get $$\label{f>}
\|f(\cdot, u_j(\cdot))\|_{L^{q/(q-1)}(\Omega)}\geq \frac{\bar \sigma}{\tilde\kappa \lambda_{k+m}}\|u_j\|_{X_0}$$ for 