Communications on Pure and Applied Analysis (CPAA)

A sharp Sobolev inequality on Riemannian manifolds

Pages: 1 - 31, Volume 2, Issue 1, March 2003      doi:10.3934/cpaa.2003.2.1

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YanYan Li - Department of Mathematics, Rutgers University, Hill Center, 110 Frelinghuysen Rd., Piscataway, NJ 08854, United States (email)
Tonia Ricciardi - Departimento di Matematica e Applicazioni, Universita di Napoli Federico II Via Cintia, 80126 Napoli, Italy (email)

Abstract: Let $(M,g)$ be a smooth compact Riemannian manifold without boundary of dimension $n\ge 6$. We prove that

$||u||_{L^{2^*}(M,g)}^2 \le K^2\int_M\{|\nabla_{g} u|^2+c(n)R_{g} u^2\}dv_g +A||u||_{L^{2n/(n+2)}(M,g)}^2,$

for all $u\in H^1(M)$, where $2^*=2n/(n-2)$, $c(n)=(n-2)/[4(n-1)]$, $R_g$ is the scalar curvature, $K^{-1}=$ inf $\|\nabla u\|_{L^2(\mathbb R^n)}\|u\|_{L^{2n/(n-2)}(\mathbb R^n)}^{-1}$ and $A>0$ is a constant depending on $(M,g)$ only. The inequality is sharp

Keywords:  sharp Sobolev inequality, critical exponent, Yamabe problem.
Mathematics Subject Classification:  35J60, 58E35.

Revised: November 2002;      Available Online: December 2002.