multiplicity result is the so-called $\nabla$-condition introduced in [@marsac:sns Definition 2.4]. This section is devoted to the verification of this condition for functional $\mathcal J_\lambda$. For this purpose, in the sequel we denote by $$P_C:X_0\to C$$ the orthogonal projection of $X_0$ onto $C$.

Let $C$ be a closed subspace of $X_0$ and $a,b\in {\mathbb R}\cup\{-\infty, +\infty\}$. We say that functional $\mathcal J_\lambda$ verifies condition $(\nabla)(\mathcal J_\lambda, C, a, b)$ if there exists $\gamma>0$ such that $$\inf\Big\{\|P_C \nabla \mathcal J_\lambda (u)\|_{X_0} : a\leq \mathcal J_\lambda(u)\leq b,\,\, dist (u, C)\leq \gamma\Big\}>0\,.$$

Roughly speaking, the condition $(\nabla)(\mathcal J_\lambda, C, a, b)$ requires that $\mathcal J_\lambda$ has no critical points $u\in C$ such that $a\leq \mathcal J_\lambda(u)\leq b$, with some uniformity. In order to prove this condition for $\mathcal J_\lambda$, we need two preliminary lemmas.

\[lemmanabla1\] Let $k$ and $m$ in ${\mathbb N}$ be such that $\lambda_{k-1}<\lambda_k=\dots = \lambda_{k+m-1}<\lambda_{k+m}$ and let $f$ be a function satisfying conditions –.

Then, for any $\sigma>0$ there exists $\varepsilon_\sigma>0$ such that for any $\lambda\in [\lambda_{k-1}+\sigma, \lambda_{k+m}-\sigma]$ the unique critical point $u$ of $\mathcal J_\lambda$ constrained on $\mathbb H_{k-1}\oplus \mathbb P_{k+m-1}$ and with $\mathcal J_\lambda(u)\in [-\varepsilon_\sigma, \varepsilon_\sigma]$, is the trivial one.

We argue by contradiction and we suppose that 