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Paneitz-type operators and applications. (English) Zbl 0998.58009
Let \((M,g)\) be a Riemannian manifold of dimension \(n\geq 5\). The authors deal with fourth-order operators \(P_g\) defined by \(P_g = \Delta^2_g + \alpha \Delta_g\), where \(\Delta_g\) is the Laplacian on \((M,g)\) and \(\alpha > 0\) is a real number. Such operators are closely related to the Paneitz operator \(\Delta^2_g - \text{div}_g(\frac{2}{3}S_gg - \text{Ric}_g)du\). Here \(S_g\) is the scalar curvature of \((M,g)\).
The authors study \(P_g\) on the Sobolev space \(H_2^2(M)\), the completion of \(C^\infty(M)\) with respect to the norm \(\|u\|^2=\|\nabla^2 u\|_2^2+\|\nabla u\|_2^2+\|u \|_2^2\). The main content of the paper is the construction of a kind of best possible Sobolev constant. To be more precise, put \(2^\sharp= \frac{2n}{n-4}\); then \[ \|u\|^2_{2^\sharp} \leq A\int_M (P_gu)u dv_g + B\|u\|^2_2 \tag{1} \] and \[ \|u\|^2_{2^\sharp} \leq A\int_M (P_gu)u dv_g + B\|u\|^2_{H_1^2}.\tag{2} \] where the last norm is the usual norm of \(H_1^2(M)\). Here the authors study the problem of finding optimal constants \(A^{(i)}_{\text{opt}}\) [resp. \(B^{(i)}_{\text{opt}}\)] for these inequalities in the case of \(u\in H_2^2(M)\) in the following sense: \(A^{(i)}_{\text{opt}} = \inf\{A\in\mathbb{R}\) such that \(\exists B\in\mathbb{R}\) with the property that \((\text{S}i)\) is valid} and analogously for \(B^{(i)}_{\text{opt}}\). The result is a sharp constant \(A^{(i)}_{\text{opt}} = K_0 = \pi^2n(n-4)(n^2-4)\Gamma ( \frac{n} {2})^{4/n}\Gamma(n)^{-4/n}\), \(i= 1,2\). The question of whether there exists a function \(u\) such that these optimal constants are attained is partially solved. If \(g\) is a conformally flat metric the constant \(A_{\text{opt}}^{(2)}\) is attained in (S2). The results are applied to the fourth-order differential equation \(P_g u + \alpha u = fu^{2^\sharp-1}\). Here the authors obtain a sufficient condition for the existence of a smooth positive solution.
The paper is rather technical; however, this will be unavoidable in the subject.

58E35 Variational inequalities (global problems) in infinite-dimensional spaces
35A15 Variational methods applied to PDEs
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
Full Text: DOI
[1] M. T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds , Invent. Math. 102 (1990), 429–445. · Zbl 0711.53038
[2] W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality , Ann. of Math. (2) 138 (1993), 213–242. JSTOR: · Zbl 0826.58042
[3] T. P. Branson, Group representations arising from Lorentz conformal geometry , J. Funct. Anal. 74 (1987), 199–291. · Zbl 0643.58036
[4] T. P. Branson, S.-Y. A. Chang, and P. C. Yang, Estimates and extremals for zeta function determinants on four-manifolds , Comm. Math. Phys. 149 (1992), 241–262. · Zbl 0761.58053
[5] S.-Y. A. Chang, On a fourth order PDE in conformal geometry , preprint, 1997.
[6] S.-Y. A. Chang, M. J. Gursky, and P. C. Yang, Regularity of a fourth order nonlinear PDE with critical exponent , Amer J. Math. 121 (1999), 215–257. · Zbl 0921.35032
[7] S.-Y. A. Chang and P. C. Yang, Extremal metrics of zeta function determinants on \(4\)-manifolds , Ann. of Math. (2) 142 (1995), 171–212. JSTOR: · Zbl 0842.58011
[8] F. Demengel and E. Hebey, On some nonlinear equations involving the \(p\)-Laplacian with critical Sobolev growth , Adv. Differential Equations 3 (1998), 533–574. · Zbl 0955.35031
[9] Z. Djadli, E. Hebey, and M. Ledoux, Sharp inequalities involving Paneitz-type operators , preprint, 1999. · Zbl 0998.58009
[10] D. E. Edmunds, F. Fortunato, and E. Janelli, Critical exponents, critical dimensions, and the biharmonic operator , Arch. Rational Mech. Anal. 112 (1990), 269–289. · Zbl 0724.35044
[11] F. Escobar and R. Schoen, Conformal metrics with prescribed scalar curvature , Invent. Math. 86 (1986), 243–254. · Zbl 0628.53041
[12] M. J. Gursky, The Weyl functional, de Rham cohomology, and Kähler-Einstein metrics , Ann. of Math. (2) 148 (1998), 315–337. JSTOR: · Zbl 0949.53025
[13] E. Hebey, Changements de métriques conformes sur la sphère: Le problème de Nirenberg , Bull. Sci. Math. (2) 114 (1990), 215–242. · Zbl 0713.53023
[14] ——–, Sobolev Spaces on Riemannian Manifolds , Lecture Notes in Math. 1635 , Springer, Berlin, 1996.
[15] ——–, Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities , Courant Lect. Notes Math. 5 , Courant Institute of Mathematical Sciences, New York, 1999.
[16] S. Ilias, Constantes explicites pour les inégalités de Sobolev sur les variétés riemanniennes compactes , Ann. Inst. Fourier (Grenoble) 33 (1983), 151–165. · Zbl 0528.53040
[17] J. L. Kazdan and F. W. Warner, Scalar curvature and conformal deformation of Riemannian structure , J. Differential Geom. 10 (1975), 113–134. · Zbl 0296.53037
[18] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities , Ann. of Math. (2) 118 (1983), 349–374. JSTOR: · Zbl 0527.42011
[19] P.-L. Lions, The concentration-compactness principle in the calculus of variations: The limit case, I , Rev. Mat. Iberoamericana 1 (1985), no. 1, 145–201.; II , (1985), no. 2, 45–121. · Zbl 0704.49006
[20] J. Moser, “On a nonlinear problem in differential geometry” in Dynamical Systems (Salvador, Brazil, 1971), Academic Press, New York, 1973, 273–280. · Zbl 0275.53027
[21] S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds , preprint, 1983.
[22] P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators , J. Math. Pures Appl. (9) 69 (1990), 55–83. · Zbl 0717.35032
[23] R. C. A. M. van der Vorst, Best constants for the embedding of the space \(H^2\cap H_0^1(\Omega)\) into \(L^2N/(N-4)(\Omega)\) , Differential Integral Equations 6 (1993), 259–276. · Zbl 0801.46033
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