Let $\Omega\subset\mathbb{R}^n$ be an open bounded domain. Let $$F:u\in H^1_0(\Omega)\mapsto \frac12\int_{\Omega} |\nabla u|^2 dx-\frac{\lambda}{2}\int_{\Omega} |u|^2 dx -\frac{1}{2^*}\int_{\Omega} |u|^{2^*} dx,$$ where $\lambda\in\mathbb{R}$ and $2^*=\frac{2n}{n-2}$ denotes the critical Sobolev exponent.
As an exercise, I have to prove that fixed $u\in H^1_0(\Omega), u\neq 0$ $$\max_{t} F(tu) =\frac1n\left(\frac{\|u\|^2 -\lambda|u|_2^2}{|u|_{2^*}^2}\right)^{\frac{n}{2}},$$ where $\|u\|^2 =\int_{\Omega} |\nabla u|^2 dx$ denotes the norm in $H^1_0(\Omega)$ and $|u|_r^r =\int_{\Omega} |u|^r dx$ the norm in the Lebesgue space $L^r(\Omega)$.
I am in truble about proving this thing. Firstly, I observed that $$F(u) =\frac12 \|u\|^2 -\frac{\lambda}{2} |u|_2^2-\frac{1}{2^*}|u|_{2^*}^{2^*}.$$ I am pretty sure that somewhere the embedding $H^1_0(\Omega)\hookrightarrow L^{2^*}(\Omega)$ is needed and I observed that $\left(\frac12 -\frac{1}{2^*}\right)=\frac1n$, so that $\frac1n$ which appears could be something like that.
Honestly, I don't know how to proceed. Could someone please help?
Thank you in advance!
${\bf EDIT.}$ To be more precise with notations: it is $$|u|_{2^*}^2 =\left(\int_{\Omega} |u|^{2^*}dx\right)^{\frac{2}{2^*}}.$$