I'm reading Dr.Luca Martinazzi's paper Existence of solutions to a higher dimensional mean-field equation on manifolds which proves that for $m \geq 1$, there is an existence result for the equation $$ \left(-\Delta_{g}\right)^{m} u+\lambda=\lambda \frac{e^{2 m u}}{\int_{M} e^{2 m u} d \mu_{g}} $$ on a closed Riemannian manifold $(M, g)$ of dimension $2 m$ for certain values of $\lambda$.
This is a little strange for me, when dimension becomes higher, we need the $m$'s power of $\Delta$:$\left(-\Delta_{g}\right)^{m} $ to obtain the existence of the solution.
What happens when we just consider the equation $$ -\Delta_{g} u+\lambda=\lambda \frac{e^{2 u}}{\int_{M} e^{2 u} d \mu_{g}} $$ on higher dimension manifold?
I read the proof and it is the Adams-Moser-Trudinger inequality part which require the $m$ 's power of $\Delta$. I'm curious what about the equation when $m=1$ but on a high dimension manifold?
