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 $W_0^{m,p}\left(\Omega \right)$ 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.