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

Abstract

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