LAUSR

Abstract We prove that the realization A p in L p ( R N ) , 1 p ∞ , of the elliptic operator A = ( 1 + | x | α ) Δ + b | x | α − 1 x | x | ⋅ ∇ − c | x | β with domain D ( A p ) = { u ∈ W 2 , p ( R N ) | A u ∈ L p ( R N ) } 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.