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In this paper, we present a compactness embedding theorem of a subspace of a the Sobolev space W 1;p (R N ) into L s (R N ) spaces, formed by functions u(x;y) which are spherically symmetry with respect to Þrst variable and R a(z)juj p dz a0 > 0 if jzj i R; for a large R > 0 and such that lim jyj!1 a(x;y) = +1 uniformly for x: Moreover, we give two

A COMPACTNESS EMBEDDING LEMMA, A PRINCIPLE OF SYMETRIC CRITICALITY AND APPLICATIONS TO ELLIPTIC PROBLEMS