Orlicz and Sobolev spaces on unbounded domains

Abstract

This paper deals with embedding theorems involving the Sobolev spaces H m,p (Ω) and H 0 m , p ( Ω ) on an unbounded domain Ω in R n ( n > 1 ) . It is shown, for example, that the Sobolev space H 0 1 , n ( Ω ) is continuously embedded in the Orlicz space L Φ* (Ω), where Φ ( t ) = | t | n exp ⁡ ( | t | n / ( n − 1 ) ) ; and that multiplication by suitable functions acts as a compact map of H 0 1 , n ( Ω ) to L Ψ ∗ ( Ω ) for any Orlicz function Ψ subordinate to Φ in a certain sense. These results extend the earlier work of Trudinger, who dealt with the case in which Ω is bounded. Examples are given of unbounded domains Ω for which the natural embedding of in H 1 , p ( Ω ) in L p ( Ω ) ( 1 < p < ∞ ) is a k -set contraction for some k < 1: the case k = 0 corresponds to a compact embedding. Applications are made to the Dirichlet problem in an unbounded domain for elliptic equations with violent non-linearities.