What does it mean that an equation is conformally invariant? I'am reading a PDE paper which says that the following critical Yamabe equation is conformally invariant $$-\Delta u = \frac{n(n-2)}{4}|u|^{\frac{4}{n-2}}u,\quad u\in\mathcal{D}^{1,2}(\mathbb{R}^n),$$ where $\mathcal{D}^{1,2}(\mathbb{R}^n)$ is the closure of $C_0^{\infty}(\mathbb{R}^n)$ under the norm $\|\nabla u\|_{L^2}$.
I want to know the meaning of conformally invariant. And it will be appreciated for any reference about this topic. Thank you!
In this context, conformally invariant means that solutions of the PDE are invariant under a certain action of the group of Möbius transformations. Specifically, let $\Phi \colon \mathbb{R}^n \to \mathbb{R}^n$ be a Möbius transformation; i.e. a composition of isometries, dilations $x \mapsto \lambda x$, and inversions $x \mapsto \frac{x}{\lvert x\rvert}$. Define $\Phi \cdot u$ by
$$ \Phi \cdot u = \lvert J_\Phi \rvert^{\frac{n-2}{2n}} (u \circ\Phi) , $$
where $\lvert J_\Phi \rvert$ is the Jacobian determinant of $\Phi$. One can show that $u$ is a solution of your PDE if and only if $\Phi \cdot u$ is a solution for any Möbius transformation $\Phi$. This can be done by checking individually for isometries (easy), dilations (easy), and inversions (a little more work, but not too hard). Alternatively, this can be done by realizing your PDE as the Euler equation of a conformally invariant functional.
A good reference, which includes the geometric motivation for this problem, is Lee and Parker’s survey article on the Yamabe Problem.