Rigidity of manifolds admitting stable solutions of an elliptic problem

In this paper, we study geometric rigidity of Riemannian manifolds admitting stable solutions of certain elliptic problems (stability in a variational sense), that is, under suitable hypotheses, we are able to characterize the Riemannian manifold which admits a stable solution. Furthermore, under the non-negativity of the weighted Ricci curvature, we deduce several data about the stable solution and a splitting result for the manifold.