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Paneitz-type operators and applications. (English) Zbl 0998.58009

Let (M,g) be a Riemannian manifold of dimension n5. The authors deal with fourth-order operators P g defined by P g =Δ g 2 +αΔ g , where Δ g is the Laplacian on (M,g) and α>0 is a real number. Such operators are closely related to the Paneitz operator Δ g 2 -div g (2 3S g g-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 (M) with respect to the norm u 2 = 2 u 2 2 +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 =2n n-4; then

u 2 2 A M (P g u)udv g +Bu 2 2 (1)

and

u 2 2 A M (P g u)udv g +Bu H 1 2 2 ·(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 opt (i) [resp. B opt (i) ] for these inequalities in the case of uH 2 2 (M) in the following sense: A opt (i) =inf{A such that B with the property that (Si) is valid} and analogously for B opt (i) . The result is a sharp constant A opt (i) =K 0 =π 2 n(n-4)(n 2 -4)Γ(n 2) 4/n Γ(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 opt (2) is attained in (S2). The results are applied to the fourth-order differential equation P g u+αu=fu 2 -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.


MSC:
58E35Variational inequalities (global problems)
35A15Variational methods (PDE)
58J60Relations of PDE with special manifold structures