A priori estimates and Liouville type results for quasilinear elliptic equations involving gradient terms

Abstract In this article we study local and global properties of positive solutions of − Δ m u = ∣ u ∣ p −1 u + M ∣∇ u ∣ q in a domain Ω of ℝ N , with m > 1, p, q > 0 and M ∈ ℝ. Following some ideas used in [7, 8], and by using a direct Bernstein method combined with Keller–Osserman’s estimate, we obtain several a priori estimates as well as Liouville type theorems. Moreover, we prove a local Harnack inequality with the help of Serrin’s classical results.