Hardy-type inequalities for the drifting p-Laplace operator and applications

We consider the drifting p -Laplace operator {equation*} Δp,vu=e⁻ᵛ div\,(e^v |∇ u|ᵖ⁻²∇ u) {equation*} and discuss generalized weighted Hardy-type inequalities associated with the measure dμ=eᵛ⁽ˣ⁾dx . As an application, we obtain several Liouville-type results for positive solutions of the non-linear elliptic problem with singular lower order term {equation*}-Δp,v u≥ c(x) uᵖ⁻¹+B |∇ u|^p/u in\ Ω,{equation*} where Ω is a bounded or an unbounded exterior domain in RN , N > p > 1 , B+p-1 > 0 , as well as of the non-autonomous quasilinear elliptic problem {equation*}-Δp,v u+b(x)|∇ u|ᵖ⁻¹ ≥ c(x) uᵖ⁻¹ in\ Ω,{equation*} with general weights b≥0 and c > 0. Liouville-type results are also discussed for a class of higher order differential equations.