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Consider a Riemannian manifold $(M^n, g)$ of dimension $n$ with a metric $g$. We assume $M$ to be closed (compact without boundary). Let's not assume any hypothesis on the Yamabe invariant of the manifold.
A GJMS operator on $M^n$ is a differential operator of order $n$ with principal symbol $(-\Delta_g)^{(n/2)}$.

If the Yamabe invariant on $M^n$ is positive then the kernel of this operator is just the constants (for this statement cfr Gursky - "Conformal invariants and nonlinear elliptic equations", as for the Green's function in this case you can check Ndiaye - "Constant $Q$-curvature metrics in arbitrary dimension").

In general the constants on $M^n$ are just a subset of the kernel.

Does anyone know if there exist a Green function for such operator in this case?

I cannot find any references for this case. Any help?

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I believe that the answer is Yes by way of the method of the parametrix. This equation is elliptic (the principal symbol is injective, being a power of the principal symbol of the Laplacian) so the parametrix is generated by a pseudo-differential operator.

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  • $\begingroup$ Do you have any references for this case? And what about the case of a manifold with boundary? $\endgroup$
    – gin111
    Commented Mar 21, 2014 at 21:22
  • $\begingroup$ Sorry, I'm not an expert, so I don't have ready references at my finger tips. However, there are references in the linked encyclopedia article. Also, I believe that pseudo-differential operators were essentially invented for constructing parametrices of elliptic equations. So any reasonably complete text on pseudo-differential operators should cover this. I'm not completely sure how this generalizes to manifolds with boundaries, but you should be able to find some of that information once you start chasing down references. $\endgroup$
    – Igor Khavkine
    Commented Mar 21, 2014 at 21:40

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