Elliptic operators with unbounded diffusion, drift and potential terms

Abstract

We prove that the realization Ap in Lp(RN),1<p<∞, of the elliptic operator A=(1+|x|α)Δ+b|x|α−1x|x|⋅∇−c|x|β with domain D(Ap)={u∈W2,p(RN)|Au∈Lp(RN)} generates a strongly continuous analytic semigroup T(⋅) provided that α>2,β>α−2 and any constants b∈R and c>0. This generalizes the recent results in [4] and in [16]. Moreover we show that T(⋅) is consistent, immediately compact and ultracontractive.