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

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