# Link between Yamabe invariant and Yamabe equation

I am trying to understand the solution to the Yamabe problem as presented by Lee and Parker. It seems to me that the constant $$\lambda$$ which appears in the Yamabe equation $$\square\varphi = \lambda \varphi^{p-1}$$ and the Yamabe invariant $$\lambda = \inf_\varphi Q_g(\varphi)$$ where $$Q_g$$ is the functional $$Q_{g}(\varphi) = \frac{\int_{M} \left(a|\nabla \varphi|^{2}+S \varphi^{2}\right) d V_{g}}{\|\varphi\|_{p}^{2}} = \frac{E(\varphi)}{\|\varphi\|_{p}^{2}}$$ are in fact the same thing, that is to say if $$\varphi$$ is an absolute minimum for the functional $$Q_g$$ then $$\varphi$$ is a solution to the Yamabe equation with $$\lambda$$ as the coefficient of $$\varphi^{p-1}$$. It seems to me that this is proved explicitely by Aubin in his book "Some nonlinear problems in Riemannian geometry".

However, an explicit calculation of the Euler-Lagrange equation for $$Q_g$$ (which is shown on page 39 of this paper by Lee and Parker and which can be easily done explicitly)

Lee, John M.; Parker, Thomas H., The Yamabe problem, Bull. Am. Math. Soc., New Ser. 17, 37-91 (1987). ZBL0633.53062.

shows that the Euler equation is $$a \Delta \varphi+S \varphi-\|\varphi\|_{p}^{-p} E(\varphi) \varphi^{p-1}=0$$ that is to say the Yamabe equation with $$\lambda = \frac{E(\varphi)}{\|\varphi\|_{p}^{p}}.$$ For the Yamabe invariant to be the same constant which appears in the Yamabe equation, the exponent at the denominator should be 2 and not $$p$$. I am therefore a bit confused: is the Yamabe invariant the same constant in the Yamabe equation in presence of a solution $$\varphi$$? If yes, where am I going wrong?

• Note that $Q_g(\varphi)$ is invariant under rescaling of $\varphi$ by a constant, but the first version you give of the Yamabe equation is not invariant under this rescaling unless you assume $\lambda$ rescales, too. So it is probably correct only, if you normalize $\varphi$ somehow ($\|\varphi\|_p = 1$ works). – Deane Yang Sep 17 at 21:44
• In general a good sanity check when writing equations and estimates is to check that everything rescales correctly when you rescale either the function or metric. – Deane Yang Sep 17 at 21:44