Existence of solutions to a higher dimensional mean-field equation on manifolds

Mircea Petrache, Luca Martinazzi

Abstract:

For $m\geq 1$ we prove an existence result for the equation
$$(-\Delta_g)^m u+\lambda=\lambda\frac{e^{2mu}}{\int_M e^{2mu}d\mu_g}$$
on a closed Riemannian manifold $(M,g)$ of dimension $2m$ for certain values of $\lambda$.