<i>Q</i>‐Curvature on a Class of Manifolds with Dimension at Least 5

Hang, Fengbo; Yang, Paul

doi:10.1002/cpa.21623

Cited by 62 publications

(63 citation statements)

References 37 publications

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“…One key ingredient in such works is that a strong maximum principle for the fourth order Paneitz-Branson operator is discovered under a hypothesis on the positivity of some conformal invariants or Q-curvature of the background metric. The readers are referred to [8,9,10,13] and the references therein. This naturally stimulates us to study GJMS operator of order six and its associated Q-curvature problem, the analogue to the Yamabe problem and Q-curvature problem for Paneitz-Branson operator.…”

Section: Introductionmentioning

confidence: 99%

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Remarks on GJMS operator of order six

Chen

Hou

2017

Pacific J. Math.

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We study analysis aspects of the sixth order GJMS operator $P_g^6$. Under conformal normal coordinates around a point, the expansions of Green's function of $P_g^6$ with pole at this point are presented. As a starting point of the study of $P_g^6$, we manage to give some existence results of prescribed $Q$-curvature problem on Einstein manifolds. One among them is that for $n \geq 10$, let $(M^n,g)$ 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 $\tilde g$ of $g$ such that its $Q$-curvature $Q_{\tilde g}^6$ equals $f$.Comment: to appear Pacific Journal of Mathematic

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Section: Introductionmentioning

confidence: 99%

“…In section 2, the expansions of Green's function for P g when n ≥ 7 are presented under conformal normal coordinates around a point. The technique used here is basically inspired by Lee-Parker [12], see also [10]. The complicate computations of the term P g (r 6−n ) are left to Appendix A, where r is the geodesic distant from this point.…”

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confidence: 99%

Remarks on GJMS operator of order six

Chen

Hou

2017

Pacific J. Math.

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“…Compare our proof with Hang-Yang [29] where the authors introduced an equivalent maximizing problem involving an integral operator to solve a fourth-order elliptic differential equation. However, even though the basic idea is similar to ours, the motivation is completely different.…”

Section: Maximum Principlementioning

confidence: 92%

“…However, even though the basic idea is similar to ours, the motivation is completely different. In our situation, the technique was introduced to provide a suitable function space to work with and control the boundary behavior of solutions, but the authors in [29] utilized it to guarantee positivity of solutions.…”

Section: Maximum Principlementioning

confidence: 99%

Minimal energy solutions to the fractional Lane–Emden system: Existence and singularity formation

Choi

Kim

2019

Rev. Mat. Iberoam.

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This is the first of two papers which study asymptotic behavior of minimal energy solutions to the fractional Lane-Emden system in a smooth bounded domain Ωunder the assumption that the subcritical pair (p, q) approaches to the critical Sobolev hyperbola. If p = 1, the above problem is reduced to the subcritical higher-order fractional Lane-Emden equation with the Navier boundary condition (−∆) s u = u n+2s n−2s −ǫ in Ω and u = (−∆) s 2 u = 0 for 1 < s < 2.The main objective of this paper is to deduce the existence of minimal energy solutions, and to examine their (normalized) pointwise limits provided that Ω is convex. As a by-product of our study, a new approach for the existence of an extremal function for the Hardy-Littlewood-Sobolev inequality is provided.

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“…This consists the main part of the note. Our approach is motivated from [BT,HY4,HWY,R]. At first we would like to show that every minimizer must be radial symmetric and decreasing with respect to some point on S 3 .…”

Section: )mentioning

confidence: 99%