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http://dx.doi.org/10.5666/KMJ.2022.62.4.751

On the Paneitz-Branson Operator in Manifolds with Negative Yamabe Constant  

Ali, Zouaoui (Department of Mathematics, Mustapha Stambouli University)
Publication Information
Kyungpook Mathematical Journal / v.62, no.4, 2022 , pp. 751-767 More about this Journal
Abstract
This paper deals with the Paneitz-Branson operator in compact Riemannian manifolds with negative Yamabe invariant. We start off by providing a new criterion for the positivity of the Paneitz-Branson operator when the Yamabe invariant of the manifold is negative. Another result stated in this paper is about the existence of a metric on a manifold of dimension 5 such that the Paneitz-Branson operator has multiple negative eigenvalues. Finally, we provide new inequalities related to the upper bound of the mean value of the Q-curvature.
Keywords
Paneitz-Branson operator; Yamabe invariant; Q-curvature; Negative eigenvalue;
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1 K. Akutagawa, Yamabe metrics of positive scalar curvature and conformally flat manifolds, Differential Geom. Appl., 4(3)(1994), 239-258.   DOI
2 T. Aubin, Nonlinear Analysis on Manifolds, Monge-Ampre Equations, Spring-Verlag(1982).
3 M. Bekiri and M. Benalili, Nodal solutions for elliptic equation involving the GJMS operators on compact manifolds, Complex Var. Elliptic Equ., 64(12)(2019), 2105-2116.   DOI
4 M. Benalili and A. Zouaoui, Elliptic equation with critical and negative exponents involving the GJMS operator on compact Riemannian manifolds, J. Geom. Phys., 140(2019), 56-73.   DOI
5 P. Buser, Riemannsche Flachen mit Eigenwerten in (0, 1/4), Comment. Math. Helv., 52(1)(1977), 25-34.   DOI
6 Y. Canzani, R. Gover, D. Jakobson and R. Ponge, Conformal invariants from nodal sets.I.Negative eigenvalues and curvature prescription, Int. Math. Res. Not. IMRN 2014, 9(2014), 2356-2400.
7 Z. Djadli, Operateurs geometriques et geometrie conforme, Seminaire de geometrie spectrale et geometrie , Grenoble, 23(2005), 49-103.
8 Z. Djadli, E. Hebey and M. Ledoux, Paneitz type operators and applications, Duke Math. J., 104(2000), 129-169.
9 C. R. Graham, R. Jenne, L. J. Masson and G. A. J. Sparling, Conformally invariant powers of the Laplacien. I.Existence, J. London Math. Soc.(2), 46(3)(1992), 557-565.   DOI
10 M. J. Gursky, The principal eigenvalue of a conformally invariant differential operator, with an application to semilinear elliptic PDE, Comm. Math. Phy., 207(1999), 131-143.   DOI
11 M. J. Gursky and A. Malchiodi. A strong maximum principal for the Paneitz operator and a nonlocal flow for the Q-curvature, J. Eur. Math. Soc. (JEMS), 17(9)(2015), 2137-2173.   DOI
12 E. Hebey, Introduction a l'analyse non lineaire sur les varietes, Diderot editeur, Arts et Sciences(1997).
13 E. Hebey and F. Robert, Coercivity and Struwe's compactness for Paneitz type operators with constant coefficients, Calc. Var. Partial Differential Equations, 13(4)(2001), 491-517.   DOI
14 A. Juhl, Explicit formulas for GJMS-operators and Q-curvatures, Geom. Funct. Anal., 23(4)(2013), 1278-1370.   DOI
15 K. Tahri, Multiple Solutions to polyharmonic elliptic problem involving GJMS operator on compact manifolds, Afr. Mat., 31(2020), 437-454.   DOI
16 J. Lohkamp, Discontinuity of geometric expansions, Comment. Math. Helv., 71(2)(1996), 213-228.   DOI
17 S. Mazumdar, GJMS-type operators on a compact Riemannian manifold: best constants and Coron-type solutions, J. Differential Equations, 261(9)(2016), 4997-5034.   DOI
18 S. Paneitz. A quartic conformally covariant differential operator for arbitrary pseudoRiemannian manifolds (summary), SIGMA Symmetry integrability Geom. Methods Appl., 4(2008), 3pp.
19 Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc., 117(1965) 251-275.   DOI
20 A. P. C. Yang and S. Y. A. Chang, On a fourth order curvature invariant, Contemp. Math., 237(1999).
21 P. C. Yang and X. W. Xu, Positivity of Paneitz operators, Discrete Contin. Dynam. Systems, 7(2)(2001), 329-342.   DOI
22 M. Benalili and K. Tahri, Nonlinear elliptic fourth order equations existence results, Nonlinear differential equations and applications, NoDEA Nonlinear Differential Equations Appl., 18(5)(2011), 539-556.   DOI