Sharp [formula omitted]-Moser inequality on Riemannian manifolds

Abstract

We consider (M,g) a smooth compact Riemannian manifold of dimension n≥2 without boundary, 1<p a real parameter and r=p(n+p)n. This paper concerns the validity of the optimal Moser inequality(∫M|u|rdvg)τp≤(A(p,n)τp(∫M|∇gu|pdvg)τp+Bopt(∫M|u|pdvg)τp)(∫M|u|pdvg)τn.This kind of inequality was already studied in the last years in the particular cases 1<p<n. Here we solve the case n≤p and we introduce one more parameter 1≤τ≤min⁡{p,2}. Moreover, we prove the existence of an extremal function for the optimal inequality above.