dx \\
&-\langle P\big(u_j-\mathcal L_K^{-1}(\lambda
 u_j+f(x,u_j))\big), u_j\rangle_{X_0}.
\end{aligned}$$ Since $\langle Pu,v\rangle_{X_0}=\langle u,Pv\rangle_{X_0}$ for any $u,v\in X_0$, we have $$\label{add22}
\begin{aligned}
\langle P\big(u_j-\mathcal L_K^{-1}(\lambda
u_j+f(x,u_j))\big), u_j\rangle_{X_0} & =\|Pu_j\|^2_{X_0}-\lambda \langle Pu_j, \mathcal L_K^{-1}u_j\rangle_{X_0}\\
& \qquad -\langle Pu_j, \mathcal L_K^{-1}f(x,u_j)\rangle_{X_0},
\end{aligned}$$ while, by , we obtain $$\label{add33}
\begin{aligned}
\lambda \langle Pu_j, \mathcal L_K^{-1}u_j\rangle_{X_0}+\langle Pu_j, \mathcal L_K^{-1}f(x,u_j)\rangle_{X_0} & =\lambda \int_\Omega |Pu_j(x)|^2\,dx\\
& \qquad +\int_\Omega f(x,u_j(x))Pu_j(x)\,dx.
\end{aligned}$$ Therefore, by -, we get $$\label{sec}
\begin{aligned}
\langle Q\nabla \mathcal J_\lambda(u_j),u_j\rangle_{X_0}& =2\mathcal J_\lambda(u_j)+
2\int_\Omega F(x,u_j(x))\, dx -\int_\Omega f(x,u_j(x))u_j(x)\,dx\\
& \qquad -\|Pu_j\|_{X_0}^2 +\lambda \int_\Omega |Pu_j(x)|^2\, dx+\int_\Omega f(x,u_j(x))Pu_j(x)\,dx.
\end{aligned}$$

By – and we easily get that $$\label{Ffujto0}
\frac{\displaystyle{2\int_\Omega F(x,u_j(x))\, dx -\int_\Omega f(x,u_j(x))u_j(x)\,dx+\int_\Omega f(x,u_j(x))Pu_j(x)\,dx}}{\|u_j\|_{X_0}^q} \to 0$$ as $j\to +\infty$.

Now, let us show that $$\label{uinfty0}
u_\infty\equiv 0\,.$$ For this purpose, we firstly claim that $$\label{add44}
\frac{\displaystyle \int_\Omega
f(x,u_j(x))Pu_j(x)\,dx}{\|u_j\|_{X_0}^q} \to 0$$ as $j\to +\infty$. Indeed, by and , we have that a.e. $x\in \Omega$ $$\begin{aligned}
|f(x,u_j(x))Pu_j(x)| & \leq \|Pu_j\|_\infty\big(a_1+a_2|u_j(x)|^{q-1}\big)\\
&\leq \|Pu_j\|_\infty\big(a_1+a_2|\ell_q(x)|^{q-1}\big)\,,
\end{aligned}$$ while, by and the fact that all norms in $\mathbb H_{k+m-1}$ are equivalent, $$\|Pu_j\|_\infty\to 0$$ as $j\to +\infty$. Note that $Pu_j\in L^\infty(\Omega)$, since all eigenfunctions of $\mathcal L_K$ are bounded (see [@sY Proposition 2.4]). Hence, holds.

By this and we obtain that $$0\leftarrow \frac{\displaystyle{2\int_\Omega F(x,u_j(x))\, dx -\int_\Omega f(x,u_j(x))u_j(x)\,dx}}{\|u_j\|_{X_0}^q} \leq \frac{\displaystyle{(2-q)\int_\Omega 