A note on Wang's conjecture for harmonic functions with nonlinear boundary condition

We obtain some Liouville type theorems for positive harmonic functions on compact Riemannian manifolds with nonnegative Ricci curvature and strictly convex boundary and partially verifies Wang’s conjecture (J. Geom. Anal. 31 (2021)). For the specific manifold ${\mathbb{B}}^n$, we present a new proof of this conjecture, which has been resolved by Gu-Li (Math. Ann. 391(2025)). Our proof is based on a general principle of applying the P-function method to such Liouville type results. As a further application of this method, we obtain some classification results for nonnegative solutions of some semilinear elliptic equations with a nonlinear boundary condition.