# On a Fourth Order Equation in 3-D

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 8, page 1029-1042
- ISSN: 1292-8119

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topXu, Xingwang, and Yang, Paul C.. "On a Fourth Order Equation in 3-D." ESAIM: Control, Optimisation and Calculus of Variations 8 (2010): 1029-1042. <http://eudml.org/doc/90640>.

@article{Xu2010,

abstract = {
In this article we study the positivity of the 4-th order Paneitz operator
for closed 3-manifolds. We prove that the connected sum of two such
3-manifold retains the same positivity property. We also solve the
analogue of the Yamabe equation for such a manifold.
},

author = {Xu, Xingwang, Yang, Paul C.},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Paneitz operator; conformal invariance; Sobolev inequality; connected sum.; connected sum},

language = {eng},

month = {3},

pages = {1029-1042},

publisher = {EDP Sciences},

title = {On a Fourth Order Equation in 3-D},

url = {http://eudml.org/doc/90640},

volume = {8},

year = {2010},

}

TY - JOUR

AU - Xu, Xingwang

AU - Yang, Paul C.

TI - On a Fourth Order Equation in 3-D

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 8

SP - 1029

EP - 1042

AB -
In this article we study the positivity of the 4-th order Paneitz operator
for closed 3-manifolds. We prove that the connected sum of two such
3-manifold retains the same positivity property. We also solve the
analogue of the Yamabe equation for such a manifold.

LA - eng

KW - Paneitz operator; conformal invariance; Sobolev inequality; connected sum.; connected sum

UR - http://eudml.org/doc/90640

ER -

## References

top- M. Ahmedou, Z. Djadli and A. Malchiodi, Prescribing a fourth order conformal invariant on the standard sphere. Part I: Perturbation Result. Comm. Comtemporary Math. (to appear). Zbl1023.58020
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- Y.S. Choi and X. Xu, Nonlinear biharmonic equation with negative exponent. Preprint (1999).
- Z. Djadli, E. Hebey and M. Ledoux, Paneitz operators and applications. Duke Math. J.104 (2000) 129-169. Zbl0998.58009
- C. Fefferman and R. Graham, Conformal Invariants, in Élie Cartan et les Mathématiques d'aujourd'hui. Asterisque (1985) 95-116.
- E. Hebey and F. Robert, Coercivity and Struwe's compactness for Paneitz type operators with constant coefficients. Preprint. Zbl0998.58007
- S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds. Preprint (1983). Zbl1145.53053
- X. Xu and P. Yang, Positivity of Paneitz operators. Discrete Continuous Dynam. Syst.7 (2001) 329-342. Zbl1032.58018

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