In this paper, we present a compactness embedding theorem of a subspace of a the Sobolev space W1,p(RN) into Ls (RN) spaces, formed by functions u(x, y) which are spherically symmetry with respect to first variable and ʃ a(z)|u|pdz < ∞, where a is a continuous function satisfying a(z) > a0 > 0 if |z| ≥ R, for a large R > 0 and such that lim |y|→∞ a(x, y) = +∞ uniformly for x. Moreover, we give two applications to elliptic problems.