Abstract. The aim of this paper is to introduce new forms of the weak and Omori-Yau maximum principles for linear operators, notably for trace type operators, and show their usefulness, for instance, in the context of PDE's and in the theory of hypersurfaces. In the final part of the paper we consider a large class of non-linear operators and we show that our previous results can be appropriately generalized to this case.

Abstract. Under appropriate spectral assumptions we prove two existence results for positive solutions of Lichnerowicz-type equations on complete manifolds. We also give a priori bounds and a comparison result that immediately yields uniqueness for certain classes of solutions. No curvature assumptions are involved in our analysis.

We prove for some singular kernels K(x, y) that viscosity solutions of the integro-differential equation

a b s t r a c tIn the sub-Riemannian setting of Carnot groups, this work investigates a priori estimates and Liouville type theorems for solutions of coercive, quasilinear differential inequalities of the typePrototype examples of ∆ ϕ G are the (subelliptic) p-Laplacian and the mean curvature operator. The main novelty of the present paper is that we allow a dependence on the gradient l(t) that can vanish both as t → 0 + and as t → +∞. Our results improve on the recent literature and, by means of suitable counterexamples, we show that the range of parameters in the main theorems are sharp.

We prove an existence theorem for positive solutions to Lichnerowicztype equations on complete manifolds with boundary (M, ∂M, , ) and nonlinear Neumann conditions. This kind of nonlinear problems arise quite naturally in the study of solutions for the Einstein-scalar field equations of General Relativity in the framework of the so called Conformal Method.