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

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

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