On properties of solutions of quasilinear second-order elliptic inequalities

Let ΩΩ be an unbounded open subset of RnRn, n≥2n≥2, and A:Ω×Rn→RnA:Ω×Rn→Rn be a function such that C1|ζ|p≤ζA(x,ζ),|A(x,ζ)|≤C2|ζ|p−1 with some constants C1>0C1>0, C2>0C2>0, and p>1p>1 for almost all x∈Ωx∈Ω and for all ζ∈Rnζ∈Rn. We obtain blow-up conditions and priori estimates for inequalities of the formdivA(x,Du)+b(x)|Du|α≥q(x)g(u)in  Ω, where p−1≤α≤pp−1≤α≤p is a real number and bb, qq, and gg are non-negative functions.