We study the local properties of positive solutions of the equation −∆u = u p − m |∇u| q in a punctured domain Ω \ {0} of R N or in a exterior domain R N \ B r0 in the range min{p, q} > 1 and m > 0. We prove a series of a priori estimates depending p and q, and of the sign of q − 2p p+1 and q − p. Using various techniques we obtain removability results for singular sets and we give a precise description of behaviour of solutions near an isolated singularity or at infinity in R N .