A note on the strong maximum principle and the compact support principle

In this note we are concerned with the strong maximum principle (SMP) and the compact support principle (CSP) for non-negative solutions to quasilinear elliptic inequalities of the formdiv(A(|∇u|)∇u)+G(|∇u|)−f(u)⩽0in Ω, anddiv(A(|∇u|)∇u)+G(|∇u|)−f(u)⩾0in RN∖Br(0), respectively. We give new conditions on the data (A,G,f)(A,G,f) to obtain (SMP) and (CSP). When these conditions are particularized to the m-Laplacian and pure power nonlinearities we completely classify the data according to the validity of the (CSP) or the (SMP). In doing so we clarify the general situation and we consider a case not covered in the literature.