Extremals for sharp GNS inequalities on compact manifolds

Autor: Marcos Montenegro, Jurandir Ceccon, Emerson Abreu
Rok vydání: 2014
Předmět:
Zdroj: Annali di Matematica Pura ed Applicata (1923 -). 194:1393-1421
ISSN: 1618-1891
0373-3114
DOI: 10.1007/s10231-014-0426-2
Popis: Let $$(M,g)$$ be a closed Riemannian manifold of dimension $$n \ge 2$$ . In Ceccon and Montenegro (Math Z 258:851–873, 2008; J Diff Equ 254(6):2532–2555, 2013) showed that, for any $$1 < p \le 2$$ and $$1 \le q < r < p^* = \frac{np}{n-p}$$ , there exists a constant $$B$$ such that the sharp Gagliardo–Nirenberg inequality $$\begin{aligned} \left( \int _M |u|^r\; \mathrm{d}v_g \right) ^{\frac{p}{r \theta }} \le \left( A_{\mathrm{opt}} \int _M |\nabla _g u|^p\; \mathrm{d}v_g + B \int _M |u|^p\; \mathrm{d}v_g \right) \left( \int _M |u|^q\; \mathrm{d}v_g \right) ^{\frac{p(1 - \theta )}{\theta q}}. \end{aligned}$$ holds for all $$u \in C^\infty (M)$$ . In this work, assuming further $$1 < p < 2, p < r$$ and $$1 \le q \le \frac{r}{r-p}$$ , we derive existence and compactness results of extremal functions corresponding to the saturated version of the above sharp inequality. Sobolev inequality can be seen as a limiting case as $$r$$ tends to $$p^*$$ .
Databáze: OpenAIRE
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