A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics

In this paper we analyze the existence and non-existence of cylindrical solutions for a nonlinear elliptic equation in R-3, which has been proposed as a model for the dynamics of galaxies. We prove a general integral inequality of Sobolev-Hardy type that allows us to use variational methods when the power p belongs to the interval [4, 6]. We find solutions in the range 4 < p less than or equal to 6. The value p = 4 seems to have characteristics similar to those of the critical Sobolev exponent p = 6.