$$\label{J'00}
\sup\Big\{ \big|\langle\,\mathcal J'_\lambda(u_j),\varphi\,\rangle \big|\,: \;
\varphi\in
X_0\,, \|\varphi\|_{X_0}=1\Big\}\to 0$$ as $j\to +\infty$.

\[lemmaPS\] Let $\lambda>0$ and $f$ be a function satisfying conditions –.

Then, functional $\mathcal J_\lambda$ satisfies the Palais–Smale condition at any level $c\in {\mathbb R}$.

Let $c\in {\mathbb R}$ and let $\{u_j\}_{j\in {\mathbb N}}$ be a Palais–Smale sequence for $\mathcal J_\lambda$ at level $c$. First of all, let us prove that $$\label{ujlimitata}
\mbox{the sequence}\,\,\, \{u_j\}_{j\in {\mathbb N}} \,\,\, \mbox{is bounded in} \,\,\, X_0\,.$$ For this purpose, we note that by and for any $j\in {\mathbb N}$ it easily follows that there exists $\kappa>0$ such that $$\Big|\langle \mathcal J'_\lambda(u_j), \frac{u_j}{\|u_j\|_{X_0}}\rangle\Big| \leq \kappa\,,$$ and $$|\mathcal J_\lambda(u_j)|\leq \kappa\,,$$ so that, as a consequence of these two relations, we also have $$\label{kappaPS}
\mathcal J_\lambda(u_j)-\frac 1 \mu \langle \mathcal J'_\lambda(u_j), u_j\rangle\leq \kappa \left(1+ \|u_j\|_{X_0}\right)\,,$$ where $\mu$ is a parameter such that $\mu\in (2, q)$.

Now, thanks to and we get $$\label{jj'0L}
\begin{aligned}
\mathcal J_\lambda(u_j)-\frac 1 \mu \langle \mathcal J_\lambda'(u_j), u_j\rangle & = \left(\frac 1 2 -\frac 1 \mu\right)\Big(\|u_j\|_{X_0}^2-\lambda \|u_j\|_{L^2(\Omega)}^2\Big)\\
& \qquad  -\int_\Omega \Big(F(x, u_j(x))-\frac 1 \mu\, f(x, u_j(x)) \,u_j(x)\Big)\,dx\\
& \geq \left(\frac 1 2 -\frac 1 \mu\right)\Big(\|u_j\|_{X_0}^2-\lambda \|u_j\|_{L^2(\Omega)}^2\Big)\\
& \qquad +\left(\frac q \mu-1\right)\int_\Omega F(x, u_j(x))\,dx\\
& \geq \left(\frac 1 2 -\frac 1 