zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Paneitz-type operators and applications. (English) Zbl 0998.58009

Let $\left(M,g\right)$ be a Riemannian manifold of dimension $n\ge 5$. The authors deal with fourth-order operators ${P}_{g}$ defined by ${P}_{g}={{\Delta }}_{g}^{2}+\alpha {{\Delta }}_{g}$, where ${{\Delta }}_{g}$ is the Laplacian on $\left(M,g\right)$ and $\alpha >0$ is a real number. Such operators are closely related to the Paneitz operator ${{\Delta }}_{g}^{2}-{\text{div}}_{g}\left(\frac{2}{3}{S}_{g}g-{\text{Ric}}_{g}\right)du$. Here ${S}_{g}$ is the scalar curvature of $\left(M,g\right)$.

The authors study ${P}_{g}$ on the Sobolev space ${H}_{2}^{2}\left(M\right)$, the completion of ${C}^{\infty }\left(M\right)$ with respect to the norm ${\parallel u\parallel }^{2}=\parallel {\nabla }^{2}{u\parallel }_{2}^{2}+{\parallel \nabla u\parallel }_{2}^{2}+{\parallel u\parallel }_{2}^{2}$. The main content of the paper is the construction of a kind of best possible Sobolev constant. To be more precise, put ${2}^{♯}=\frac{2n}{n-4}$; then

${\parallel u\parallel }_{{2}^{♯}}^{2}\le A{\int }_{M}\left({P}_{g}u\right)ud{v}_{g}+B{\parallel u\parallel }_{2}^{2}\phantom{\rule{2.em}{0ex}}\left(1\right)$

and

${\parallel u\parallel }_{{2}^{♯}}^{2}\le A{\int }_{M}\left({P}_{g}u\right)ud{v}_{g}+B{\parallel u\parallel }_{{H}_{1}^{2}}^{2}·\phantom{\rule{2.em}{0ex}}\left(2\right)$

where the last norm is the usual norm of ${H}_{1}^{2}\left(M\right)$. Here the authors study the problem of finding optimal constants ${A}_{\text{opt}}^{\left(i\right)}$ [resp. ${B}_{\text{opt}}^{\left(i\right)}$] for these inequalities in the case of $u\in {H}_{2}^{2}\left(M\right)$ in the following sense: ${A}_{\text{opt}}^{\left(i\right)}=inf\left\{A\in ℝ$ such that $\exists B\in ℝ$ with the property that $\left(\text{S}i\right)$ is valid} and analogously for ${B}_{\text{opt}}^{\left(i\right)}$. The result is a sharp constant ${A}_{\text{opt}}^{\left(i\right)}={K}_{0}={\pi }^{2}n\left(n-4\right)\left({n}^{2}-4\right){\Gamma }{\left(\frac{n}{2}\right)}^{4/n}{\Gamma }{\left(n\right)}^{-4/n}$, $i=1,2$. 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 ${A}_{\text{opt}}^{\left(2\right)}$ is attained in (S2). The results are applied to the fourth-order differential equation ${P}_{g}u+\alpha u=f{u}^{{2}^{♯}-1}$. 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:
 58E35 Variational inequalities (global problems) 35A15 Variational methods (PDE) 58J60 Relations of PDE with special manifold structures