In this paper, we consider two overdetermined problems for the anisotropic $p$-Laplacian ( $1<p\le n$) in the exterior domains and the bounded punctured domains, respectively, and prove the corresponding Wulff shape characterizations, by using Weinberger type approach.

It was conjectured by Escobar (J Funct Anal 165:101–116, 1999) that for an ndimensional (n ≥ 3) smooth compact Riemannian manifold with boundary, which has nonnegative Ricci curvature and boundary principal curvatures bounded below by c > 0, the first nonzero Steklov eigenvalue is greater than or equal to c with equality holding only on isometrically Euclidean balls with radius 1/c. In this paper, we confirm this conjecture in the case of nonnegative sectional curvature. The proof is based on a combination of Qiu–Xia’s weighted Reilly-type formula with a special choice of the weight function depending on the distance function to the boundary, as well as a generalized Pohozaev-type identity.