we know that the first eigenvalue of Laplacian on the  Riemannian unit shpere $S^n$ is $n$, then what is the explicit expression for the first eigenvalue of $p$-Lapalcian on $S^n$? 

The $p$-Laplacian eigevalue equation is 
$-div(|\nabla u|^{p-2}\nabla u)=\lambda| u|^{p-2}u$,   for any $p>1$.