Abstract:

Let $(M^n,g)$ be a $n$-dimensional compact Riemannian manifolds, $0<\alpha<n$ and $q>\frac{n}{n-\alpha}$. This paper is mainly devoted to study the extremal problems of the following HLS inequalities:

$\|I_\alpha f\|_{L^q(M^n)}\leq C\|f\|_{L^p(M^n)},\quad\mbox{with}\quad I_\alpha f(x)=\int_{M^n}\frac{f(y)}{|x-y|_g^{n-\alpha}}{\rm d}V_y,\ p\geq\frac{nq}{n+\alpha q}.$

Firstly, we prove that $I_\alpha: L^p(M^n)\rightarrow L^q(M^n)$ with $p>\frac{nq}{n+\alpha q}$ is compact and then get the existence of extremal functions $f_p, p>\frac{nq}{n+\alpha q}$. Secondly, we find that the function sequence $\{f_p\}$ is a maximizing sequence for the sharp constant of HLS inequality with $p=\frac{nq}{n+\alpha q}$. Finally, by the Concentration-Compactness principle established in [32], we can prove that there exists a convergence subsequence of $\{f_p\}$ and then give the existence of extremal function for critical case.

Key words: Hardy-Littlewood-Sobolev intqualities, Compact Riemannian manifolds, Extremal problems