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

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