caa a22 4500 605496536 CHVBK 20210128100536.0 cr unu---uuuuu 210128e20151001xx s 000 0 eng 10.1007/s10231-014-0426-2 doi (NATIONALLICENCE)springer-10.1007/s10231-014-0426-2 Extremals for sharp GNS inequalities on compact manifolds [Elektronische Daten] [Emerson Abreu, Jurandir Ceccon, Marcos Montenegro] Let $$(M,g)$$ ( M , g ) be a closed Riemannian manifold of dimension $$n \ge 2$$ n ≥ 2 . In Ceccon and Montenegro (Math Z 258:851-873, 2008; J Diff Equ 254(6):2532-2555, 2013) showed that, for any $$1 < p \le 2$$ 1 < p ≤ 2 and $$1 \le q < r < p^* = \frac{np}{n-p}$$ 1 ≤ q < r < p ∗ = n p n - p , there exists a constant $$B$$ B such that the sharp Gagliardo-Nirenberg inequality $$\begin{aligned} \left( \int _M |u|^r\; \mathrm{d}v_g \right) ^{\frac{p}{r \theta }} \le \left( A_{\mathrm{opt}} \int _M |\nabla _g u|^p\; \mathrm{d}v_g + B \int _M |u|^p\; \mathrm{d}v_g \right) \left( \int _M |u|^q\; \mathrm{d}v_g \right) ^{\frac{p(1 - \theta )}{\theta q}}. \end{aligned}$$ ∫ M | u | r d v g p r θ ≤ A opt ∫ M | ∇ g u | p d v g + B ∫ M | u | p d v g ∫ M | u | q d v g p ( 1 - θ ) θ q . holds for all $$u \in C^\infty (M)$$ u ∈ C ∞ ( M ) . In this work, assuming further $$1 < p < 2, p < r$$ 1 < p < 2 , p < r and $$1 \le q \le \frac{r}{r-p}$$ 1 ≤ q ≤ 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 $$r$$ r tends to $$p^*$$ p ∗ . Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg, 2014 Sharp Sobolev inequalities nationallicence De Giorgi-Nash-Moser estimates nationallicence Extremal functions nationallicence Compactness nationallicence Abreu Emerson Departamento de Matemática, Universidade Federal de Minas Gerais, Caixa Postal 702, 30123-970, Belo Horizonte, MG, Brazil aut Ceccon Jurandir Departamento de Matemática, Universidade Federal do Paraná, Caixa Postal 019081, 81531-990, Curitiba, PR, Brazil aut Montenegro Marcos Departamento de Matemática, Universidade Federal de Minas Gerais, Caixa Postal 702, 30123-970, Belo Horizonte, MG, Brazil aut Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/5(2015-10-01), 1393-1421 0373-3114 194:5<1393 2015 194 10231 https://doi.org/10.1007/s10231-014-0426-2 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10231-014-0426-2 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Abreu Emerson Departamento de Matemática, Universidade Federal de Minas Gerais, Caixa Postal 702, 30123-970, Belo Horizonte, MG, Brazil aut NATIONALLICENCE 700 1- Ceccon Jurandir Departamento de Matemática, Universidade Federal do Paraná, Caixa Postal 019081, 81531-990, Curitiba, PR, Brazil aut NATIONALLICENCE 700 1- Montenegro Marcos Departamento de Matemática, Universidade Federal de Minas Gerais, Caixa Postal 702, 30123-970, Belo Horizonte, MG, Brazil aut NATIONALLICENCE 773 0- Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/5(2015-10-01), 1393-1421 0373-3114 194:5<1393 2015 194 10231