Suppose we have an n-manifold ($M$, $g_0$) with scalar curvature $S_0$. I'm aware that the Yamabe problem can be classed as the problem of minimising the following Einstein-Hilbert Functional over the conformal class of $g_0$.
$$\underset{g \in [g_0]}{\mathcal{E}({g})} = {\frac{{\int_{\mathcal{M}}{S} {dvol_{{g}}}}}{\left({\int_{\mathcal{M}}dvol_{{g}} }\right)^{{\frac{n-2}{n}}}}}.$$
The metric which minimises this, say $\tilde{g}$, is the metric we're looking for with constant scalar curvature.
All four authors who contributed to the proof of this problem (Yamabe, Trudinger, Aubin, and Schoen) found that this is equivalent to minimising the following functional for $u$.
$$F(u) = {\frac{{\int_{\mathcal{M}}\frac{4(n-1)}{n-2}} {\lvert \nabla u \rvert}^2 + S_0 u^2}{{\| u \|}_p^2}}.$$ and $p=\frac{2n}{n-2}$
The minimising function $\tilde{u}$ then gives our required metric $\tilde{g} = \tilde{u}^{\frac{4}{n-2}}g_0$ which has constant scalar curvature.
Moreover, the scalar curvature of an arbitrary metric $\tilde{g}$ in $g_0$'s conformal class can be given by
$$\tilde{S} = u^{\frac{-4}{n-2}}\left(S_0 - {\frac{4(n-1)}{n-2}} u^{-1} \mathop{}\!\mathbin\bigtriangleup u\right)$$ where $\mathop{}\!\mathbin\bigtriangleup$ denotes the generalised Laplacian for manifolds.
${\bf Question:}$ How are these two variational problems equivalent? I would imagine it is a simple case of plugging the formula for $\tilde{S}$ into our formula for $\mathcal{E}(\tilde{g})$. However I am unsure how the volume forms transform under this relation?
It is a conformal transformation, which means that the volume form transforms as $$ dvol_{\tilde g} = u^{\frac{2n}{n-2}}vol_{g_0}, $$ and so $$ \int_Mdvol_{\tilde g} = \int_M u^{\frac{2n}{n-2}}vol_{g_0} = \|u\|_p^p, $$ with $p=\frac{2n}{n-2}$. By the way I feel that the transformation formula should be $\tilde g=u^{\frac{2}{n-2}}g_0$, not $\tilde g=u^{\frac{4}{n-2}}g_0$.