A Liouville result on a half space

In this paper we prove a Liovillle theorem for nonnegative weak solutions on a half space for elliptic inequalities dealing with a general class of weakly coercive operators, introduced by Farina and Serrin, which include the Laplacian operator. Neither symmetry and assumptions on the behavior of solutions at infinity nor use of comparison principles are required. We improve a previous result of Bidaut-Vèron and Pohozaev in 2002, where no explicit dependence on x and u is considered inside the divergence operator. Liouville theorems can be used to find a priori estimates and existence of solutions of Dirichlet problems on bounded domains.