Some characterizations of Bach solitons via Ricci curvature

In this short note we provide some results for Bach solitons under different assumptions. In fact, under either non-negative or non-positive Ricci curvature condition we are able to show that a Bach soliton must be Bach-flat, since it satisfies a finite weighted Dirichlet integral condition or a parabolicity condition jointly with some regularity conditions $L^{\infty }$ or $L^p$ on gradient of the potential function.