An overdetermined problem related to the p-Laplacian on Riemannian manifolds

In this paper, we study the overdetermined problem for the p-Laplacian equation on a compact Riemannian manifold with positive Ricci curvature. By introducing a new P-function which is related to the first nonzero eigenvalue for p-Laplacian, we obtain some integral identities. As their applications, the Heintze-Karcher type inequality and the Soap Bubble Theorem have been achieved.