A Trudinger-Moser inequality in a weighted Sobolev space and applications

摘要

We establish a Trudinger-Moser type inequality in a weighted Sobolev space. The inequality is applied in the study of the elliptic equation -div(K(x) {nabla}u) = K(x) f (u) + h in R~2, where K(x) = exp(|x|~2/4), f has exponential critical growth and h belongs to the dual of an appropriate function space. We prove that the problem has at least two weak solutions provided h ≠ 0 is small.