test function $\varphi=u_\infty$), and , we get $$\label{convnorma20}
\begin{aligned}
\int_{{\mathbb R}^n\times {\mathbb R}^n}|u_\infty(x)-u_\infty(y)|^2 K(x-y)\,dx\,dy & = \lambda \int_\Omega |u_\infty(x)|^2\,dx\\
& \qquad \qquad +\int_\Omega f(x, u_\infty(x))u_\infty(x)\,dx\,.
\end{aligned}$$ Hence, and give that $$\label{convnormaX00}
\|u_j\|_{X_0}\to \|u_\infty\|_{X_0}$$ as $j\to +\infty$. With this, it is easy to see that $$\begin{aligned}
\|u_j-u_\infty\|_{X_0}^2 & = \|u_j\|_{X_0}^2 + \|u_\infty\|_{X_0}^2 -2 \int_{{\mathbb R}^n\times {\mathbb R}^n} \big(u_j(x)-u_j(y)\big)\big(u_\infty(x)-u_\infty(y)\big) K(x-y)\,dx\,dy\\
& \to 2\|u_\infty\|_{X_0}^2-2\int_{{\mathbb R}^n\times {\mathbb R}^n}|u_\infty(x)-u_\infty(y)|^2 K(x-y)\,dx\,dy=0
\end{aligned}$$ as $j\to +\infty$, thanks to and . Then, the proof of Proposition \[lemmaPS\] is complete.

Geometry of the $\nabla$-theorem {#sec:geometry}
================================

In this section we check that functional $\mathcal J_\lambda$ has the geometric structure required by the $\nabla$-theorem stated in Theorem \[thmarinosaccon\] (see Section \[sec:proofthmain\]). Precisely, we want to show that if there exist $k$ and $m$ in ${\mathbb N}$ such that $$\lambda_{k-1}<\lambda<\lambda_k=\dots = \lambda_{k+m-1}<\lambda_{k+m}$$ and $\lambda$ is sufficiently close to $\lambda_k$, then functional $\mathcal J_\lambda$ agrees with the geometric framework of Theorem \[thmarinosaccon\], taking $$\begin{aligned}
& X_1:=\mathbb H_{k-1}\\
& X_2:=\mbox{span}\left\{e_k,\ldots,e_{k+m-1}\right\}\\
& X_3:=\mathbb P_{k+m-1}\,.
\end{aligned}$$

\[propgeometria\] Let $k$ and $m$ in ${\mathbb N}$ be such that $\lambda_{k-1}<\lambda<\lambda_k=\dots = \lambda_{k+m-1}<\lambda_{k+m}$ and let $f$ be a function satisfying conditions –.

Then, there exist $\rho, R$, with $R>\rho>0$, such that $${\displaystyle \sup_{\{u\in X_1, \|u\|_{X_0}\leq R\} \cup \{u\in 