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

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