The Yamabe problem and $\phi^4$ scalar field theory? The other day I happened to be browsing this page on wikipedia: https://en.wikipedia.org/wiki/Mass_gap
In the middle of the page is the equation $$\square\phi+\lambda\phi^3=0$$ where $\square$ is the spacetime Laplacian. This is the Euler-Lagrange equation for a scalar field theory with a quartic potential.
By coincidence, I also happened to be thinking about the Yamabe problem in dimension 4, and was browsing Parker and Lee's article on the subject http://www.ams.org/journals/bull/1987-17-01/S0273-0979-1987-15514-5/S0273-0979-1987-15514-5.pdf
If you plug n=4 into equation (1.2) on page 2 of that article, you get $$(6\Delta+R)\phi=\lambda\phi^3$$
where $\Delta$ is the Laplacian with positive spectrum. This is the equation that must be satisfied by a conformal change to a metric with constant scalar curvature. At a glance it is clear that these equations are basically the same.
I tried Googling to see if there was any literature on this similarity. I assume someone else must have noticed this before, but couldn't find anything. I know that the Yamabe problem is a fundamental classical problem in Riemannian geometry, and that $\phi^4$ field theory is fundamental to physics (Witten, writing for mathematicians, discusses it a bit on page 382 here http://www.ams.org/journals/bull/2007-44-03/S0273-0979-07-01167-6/S0273-0979-07-01167-6.pdf) so it doesn't necessarily strike me as idle to wonder about a connection. Is anyone aware of whether there is anything to this observation? Is it just a coincidence? Or does it suggest some physical model associated with the Yamabe problem? (Hopefully you agree that the connection is suggestive enough to overlook how ill-drawn my questions are.)
 A: Although I cannot give an answer, I can give some comments as a physicist:
1) One can study the Euclidean version of $\phi^4$ theory, then the operator is indeed elliptic.
2) $\phi^4$ is really special in 4-dimensions. It is a marginal interaction, while anything above like $\phi^6$ is irrelevant according to RG classification and anything below is relevant. The theory can flow to an interacting conformal IR fixed point with non-zero $\lambda$, see Wilson-Fisher fixed point.
3) This equation appears also in Euclidean solutions of conformally invariant equations such as the Yang-Mills equations. In particular Merons and Instantons can be derived from a superpotential that satisfies this equation. See for example https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.51.461 (page 506)
4) One could study again these objects in geometries with negative or positive curvature, then it gets corrected with the appropriate $\, R \phi \,$ term.
5) I think conformal symmetry is the underlying connection but this is just  speculation. 
6) If you happen to know any reference with explicit solutions to this equation  would be happy to know! 
