Let $k$ and $m$ in ${\mathbb N}$ be such that $\lambda_{k-1}<\lambda_k=\dots = \lambda_{k+m-1}<\lambda_{k+m}$, let $\lambda \in {\mathbb R}$ and $f$ be a function satisfying conditions –. Moreover, let $\{u_j\}_{j\in {\mathbb N}}$ be a sequence in $X_0$ such that $$\label{condlemma1}
\{\mathcal J_\lambda(u_j)\}_{j\in {\mathbb N}}\,\,\, \mbox{is bounded in}\,\,\, {\mathbb R}\,,$$ $$\label{condlemma2}
P_{\footnotesize{\rm{span}}\left\{e_k,\ldots,\,e_{k+m-1}\right\}} u_j\to 0\,\,\, \mbox{in}\,\,\, X_0$$ and $$\label{condlemma3}
P_{\mathbb H_{k-1}\oplus \mathbb P_{k+m-1}}\nabla \mathcal J_\lambda(u_j)\to 0\,\,\, \mbox{in}\,\,\, X_0$$ as $j\to +\infty$.

Then, $\{u_j\}_{j\in {\mathbb N}}$ is bounded in $X_0$.

Assume by contradiction that $\{u_j\}_{j\in {\mathbb N}}$ is unbounded in $X_0$; without loss of generality, we can assume that $$\label{ujtoinfty}
\|u_j\|_{X_0}\to +\infty$$ as $j\to +\infty$ and that there exists $u_\infty\in X_0$ such that $$\label{convnorm}
\begin{aligned}
& \frac{u_j}{\|u_j\|_{X_0}}\rightharpoonup u_\infty\,\,\, \mbox{ in }X_0\\
& \frac{u_j}{\|u_j\|_{X_0}}\to u_\infty\,\,\, \mbox{ in }L^\nu(\Omega) \mbox{ for any $\nu\in[1,2^*)$}
\end{aligned}$$ as $j\to +\infty$ and for any $\nu\in[1,2^*)$ there exists $\ell_\nu\in L^\nu({\mathbb R}^n)$ such that $$\label{dominata20nablaadd}
|u_j(x)|\leq \ell_\nu(x) \quad \mbox{a.e. in}\,\, {\mathbb R}^n\,\quad \mbox{for any}\,\,j\in {\mathbb N}\,.$$

Now, for simplicity, we set $P_{\footnotesize{\mbox{span}}\left\{e_k,\ldots,\,e_{k+m-1}\right\}}=:P$ and $P_{\mathbb H_{k-1}\oplus \mathbb P_{k+m-1}}=:Q$, and write $$u_j=Pu_j+Qu_j\,,$$ where $Pu_j\to 0$ as $j\to\infty$ (see ). Recalling and , we have $$\label{add11}
\begin{aligned}
\langle Q\nabla \mathcal J_\lambda(u_j),u_j\rangle_{X_0}&= \langle \nabla
\mathcal J_\lambda(u_j),u_j\rangle_{X_0} - \langle P\nabla
\mathcal J_\lambda(u_j),u_j\rangle_{X_0}\\
&=\|u_j\|_{X_0}^2-\lambda \int_\Omega |u_j(x)|^2\,dx
-\int_\Omega f(x,u_j(x))u_j(x)\, 