Positive supersolutions of non-autonomous quasilinear elliptic equations with mixed reaction

We provide a simple method for obtaining new Liouville-type theorems positive supersolutions of the elliptic problem -Δ p u+b(x)|∇u| pq q+1 =c(x)u q in Ω, where Ω is an exterior domain ℝ N with N≥p>1 and q≥p-1. In case q≠p-1, we mainly deal potentials type b(x)=|x| , c(x)=λ|x| σ λ>0 a,σ∈ℝ. show that do not exist some ranges parameters p,q,a,σ, which turn out to be optimal. When q=p-1, consider above general weights b(x)≥0, c(x)>0 assume c(x)-b (x) >0 large |x|, but also allow lim |x|→∞ [c(x)-b ]=0. The b c are allowed unbounded. prove if this equation has supersolution, then must satisfy related differential inequality depending on supersolution. establish sufficient conditions nonexistence relationship values τ:=lim sup |x|b(x)≤∞. A key ingredient proofs generalized Hardy-type associated p-Laplace operator.