Local estimates and Liouville Theorems for a Class of Quasilinear Inequalities

It is well known that the theory of nonlinear partial differential equations is based on a priori estimates of their solutions. These estimates play a fundamental role both in solvability theory and Liouville theorems. When considering evolution problems, these estimates can be used to prove blow-up of solutions and the dependence of blow-up time on the initial data. In this note, we consider quasilinear elliptic equa- tions and inequalities with nonlocal nonlinearities and prove local estimates and Liouville theorems.