In this paper, we present a compactness embedding theorem of a subspace of a the Sobolev space M1,P(R ) into LS(RN) spaces, formed by functions u(x, y) which are spherically symmetry with respect to first variable and f 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 a(x, y) = +∞ uniformly for x. Moreover, we give two |y|→∞ applications to elliptic problems.