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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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