Sobolev compact embeddings in unbounded domains and its applications to elliptic equations

Abstract

We give a necessary and sufficient condition for the compact embedding of the Sobolev space W0m,p(Ω) for unbounded domains Ω. Applying this condition, we can decide whether the compact embedding holds or not. We give several examples of unbounded domains Ω satisfying the compact embedding. Using our condition, we study a semilinear elliptic equation in unbounded domains and prove the existence of a positive solution and infinitely many solutions.