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We study the qualitative behavior of non-negative entire solutions of differential inequalities with gradient terms on the Heisenberg group. We focus on two classes of inequalities: ϕ u f (u)l(|∇u|)continuous functions satisfying certain monotonicity properties.The operator ϕ , called the ϕ-Laplacian, generalizes the p-Laplace operator considered by various authors in this setting. We prove some Liouville theorems introducing two new Keller-Osserman type conditions, both extending the classical one which appeared long ago in the study of the prototype differential inequality u f (u) in R m . We show sharpness of our conditions when we specialize to the p-Laplacian. While proving these results we obtain a strong maximum principle for ϕ which, to the best of our knowledge, seems to be new. Our results continue to hold, with the obvious minor modifications, also for Euclidean space.

Abstract. In this paper we introduce the notion of Einstein-type structure on a Riemannian manifold (M, g), unifying various particular cases recently studied in the literature, such as gradient Ricci solitons, Yamabe solitons and quasi-Einstein manifolds. We show that these general structures can be locally classified when the Bach tensor is null. In particular, we extend a recent result of Cao and Chen [8].

Abstract. We are concerned with nonexistence results of nonnegative weak solutions for a class of quasilinear parabolic problems with a potential on complete noncompact Riemannian manifolds. In particular, we highlight the interplay between the geometry of the underlying manifold, the power nonlinearity and the behavior of the potential at infinity.