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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.

MSC:
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.)
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