R}^n} |u(x)-u(y)|^2 K(x-y)\, dx\,dy\right)^{q/2}\,,
\end{aligned}$$ where $c$ is a suitable universal positive constant.

Choosing $\varepsilon>0$ such that $$2\varepsilon c
|\Omega|^{(2^*-2)/2^*}<\theta\left(1-\frac{\lambda}{\lambda_k}\right),$$ inequality  reads as $$\mathcal J_\lambda(u) \geq \alpha \|u\|_{X_0}^2\left(1-\kappa
\|u\|_{X_0}^{q-2}\right)\,,$$ for suitable positive constants $\alpha$ and $\kappa$.

Now, let $\rho>0$ be sufficiently small, i.e. $\rho$ such that $1-\kappa\rho^{q-2}>0$. Then, for any $u\in \mathbb P_{k-1}$ such that $\|u\|_{X_0}=\rho$ we get that $$\mathcal J_\lambda(u)\geq \alpha \rho^2(1-\kappa \rho^{q-2})>0\,,$$ so that is proved.

Now, let us show that $$\label{supnegativo}
{\displaystyle \sup_{\{u\in X_1, \|u\|_{X_0}\leq R\} \cup \{u\in X_1\oplus X_2 : \|u\|=R\}}} \mathcal J_\lambda(u)\leq 0\,.$$ First of all, let us take $u\in \mathbb H_{k-1}$. Then $$u(x)=\sum_{i=1}^{k-1}u_i e_i(x)\,,$$ with $u_i\in {\mathbb R}$, $i=1, \dots,
k-1$ and so, by and , we deduce that $$\label{J<01}
\begin{aligned}
\mathcal J_\lambda(u) & \leq \frac{\lambda_{k-1}-\lambda}{2}\int_\Omega |u(x)|^2\,dx \leq 0\,,
\end{aligned}$$ since $\lambda_{k-1}< \lambda$.

Finally, let us consider $u\in X_1\oplus X_2=\mathbb H_k$. By we have $$\mathcal J_\lambda(u)  \leq \frac 1 2\|u\|_{X_0}^2-a_3\|u\|_{L^q(\Omega)}^q+a_4|\Omega|,$$ and the claim follows recalling that $q>2$ and that $\mathbb H_k$ is a finite-dimensional subspace of $X_0$. This and give .

Then, the assertion of Proposition \[propgeometria\] comes trivially from and .

$\nabla$-condition {#sec:nablacondition}
==================

One of the main ingredient of the $\nabla$-theorem (see [@marsac:sns Theorem 2.10]) we employ in order to get our 