Sharp $L^p$-Moser inequality on Riemannian manifolds

Abstract

We consider $(M,g)$ a smooth compact Riemannian manifold of dimension $n\ge 2$ without boundary, $1<p$ a real parameter and $r=\frac{p(n+p)}{n}$. This paper concerns the validity of the optimal Moser inequality \[ \left(\int_M |u|^r\; dv_g \right)^{\fracτ{p}} \leq \left( A(p,n)^{\fracτ{p}} \left(\int_M |\nabla_g u|^p\; dv_g\right)^{\fracτ{p}} + B_{opt} \left(\int_M |u|^p\; dv_g\right)^{\fracτ{p}} \right) \left( \int_M |u|^p\; dv_g \right)^{\fracτ{n}} \; . \] This kind of inequality was already studied in the last years in the particular cases $1<p<n$. Here we solve the case $n\le p$ and we introduce one more parameter $1\le \tau \le \min \{p,2\}$. Moreover, we prove the existence of an extremal function for the optimal inequality above.

References in corpus (1)

• Sharp Lp-entropy inequalities on manifolds