F(x,u_j(x))\, dx}}{\|u_j\|_{X_0}^q}\leq 0\,,$$ as $j\to +\infty$, also thanks to . Hence, $$\frac{\displaystyle \int_\Omega F(x,u_j(x))\,
dx}{\|u_j\|_{X_0}^q}\to 0$$ as $j\to +\infty$.

As a consequence of this, and we have that $$\frac{\displaystyle \int_\Omega |u_j(x)|^q\,
dx}{\|u_j\|_{X_0}^q}\to 0,$$ as $j\to +\infty$, which yields , thanks to .

Now, by and , we get $$\frac{\mathcal J_\lambda(u_j)}{\|u_j\|_{X_0}^2}=\frac 1 2-\frac \lambda 2\frac{\displaystyle \int_\Omega |u_j(x)|^2\,
dx}{\|u_j\|_{X_0}^2}-\frac{\displaystyle \int_\Omega F(x,u_j(x))\,
dx}{\|u_j\|_{X_0}^2}\to 0,$$ which, together with (here with $\nu=2$) and , implies that $$\label{F2}
\frac{\displaystyle \int_\Omega F(x,u_j(x))\,
dx}{\|u_j\|_{X_0}^2}\to \frac 1 2$$ as $j\to +\infty$.

Hence, as a consequence of , and , there exists $C>0$ such that $$\label{q2}
\|u_j\|_{L^q(\Omega)}^q\leq C\|u_j\|_{X_0}^2 \mbox{ for every $j\in {\mathbb N}$}.$$

Now, let us show that $$\label{fPnorma2}
\frac{\displaystyle \int_\Omega f(x,u_j(x))Pu_j(x)\,dx}{\|u_j\|_{X_0}^2}\to 0$$ as $j\to +\infty$. Indeed, by and the Hölder inequality, we have $$\begin{aligned}
\frac{\displaystyle \int_\Omega |f(x,u_j(x))Pu_j(x)|\,
dx}{\|u_j\|_{X_0}^2} & \leq
\frac{\|Pu_j\|_\infty}{\|u_j\|_{X_0}^2}\left(a_1|\Omega|+a_2\int_\Omega |u_j(x)|^{q-1}\,
dx\right)\\
& \leq \|Pu_j\|_\infty
\left[\frac{a_1|\Omega|}{\|u_j\|_{X_0}^2}+\frac{a_2'}{\|u_j\|_{X_0}^{2/q}}
\left(\frac{\displaystyle\int_\Omega |u_j(x)|^q
dx}{\|u_j\|_{X_0}^2}\right)^{1-1/q}\right]\\
& \leq \|Pu_j\|_\infty
\left[\frac{a_1|\Omega|}{\|u_j\|_{X_0}^2}+\frac{a_2'C^{1-1/q}}{\|u_j\|_{X_0}^{2/q}}\right]\,,
\end{aligned}$$ thanks to . Thus, follows from this, and .

Finally, dividing both sides of by $\|u_j\|_{X_0}^2$, using – and we get $$\frac{\displaystyle{2\int_\Omega F(x,u_j(x))\, dx -\int_\Omega f(x,u_j(x))u_j(x)\,dx}}{\|u_j\|_{X_0}^2} \to 0\,,$$ which, arguing as above, yields $$\frac{\displaystyle{2\int_\Omega F(x,u_j(x))\, dx}}{\|u_j\|_{X_0}^2} \to 0\,,$$ as $j\to +\infty$. Of course, this is in contradiction with . The proof of Lemma 