Extremals for sharp GNS inequalities on compact manifolds

Abstract

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 \(p^*\).