#### Vol. 289, No. 1, 2017

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Remarks on GJMS operator of order six

### Xuezhang Chen and Fei Hou

Vol. 289 (2017), No. 1, 35–70
DOI: 10.2140/pjm.2017.289.35
##### Abstract

We study analysis aspects of the sixth-order GJMS operator ${P}_{\phantom{\rule{0.3em}{0ex}}g}^{6}$. Under conformal normal coordinates around a point, we present the expansions of Green’s function of ${P}_{\phantom{\rule{0.3em}{0ex}}g}^{6}$ with pole at this point. As a starting point of the study of ${P}_{\phantom{\rule{0.3em}{0ex}}g}^{6}$, we manage to give some existence results of the prescribed $Q$-curvature problem on Einstein manifolds. One among them is that for $n\ge 10$, let $\left({M}^{n},g\right)$ be a closed Einstein manifold of positive scalar curvature and $f$ a smooth positive function in $M$. If the Weyl tensor is nonzero at a maximum point of $f$ and $f$ satisfies a vanishing order condition at this maximum point, then there exists a conformal metric $\stackrel{̃}{g}$ of $g$ such that its $Q$-curvature ${Q}_{\stackrel{̃}{g}}^{6}$ equals $f$.

##### Keywords
sixth-order GJMS operator, prescribed $Q$-curvature problem, Green's function, mountain pass critical points
##### Mathematical Subject Classification 2010
Primary: 53A30, 53C21, 58J05
Secondary: 35B50, 35J08, 35J35