On supercritical Sobolev type inequalities and related elliptic equations

Sobolev type embeddings for radial functions into variable exponent Lebesgue spaces are studied. In particular, the following inequality is proved: let B⊂ RN, N≥ 3 , be the unit ball, and let H0,rad1(B) denote the first order Sobolev space of radial functions, and 2 ∗= 2 N/ (N- 2) the corresponding critical Sobolev embeddding exponent. Let r= |x| , and p(r) = 2 ∗+ rα, with α> 0 ; then (Formula presented.). We point out that the growth of p(r) is strictly larger than 2 ∗, except in the origin. Furthermore, we show that for p(r) = 2 ∗+ rα, with 0