Critical nonlinear elliptic equations with singularities and cylindrical symmetry

Motivated by a problem arising in astrophysics we study a nonlinear elliptic equation in $\mathbb{R}^{N}$ with cylindrical symmetry and with singularities on a whole subspace of $\mathbb{R}^{N}$. We study the problem in a variational framework and, as the nonlinearity also displays a critical behavior, we use some suitable version of the Concentration-Compactness Principle. We obtain several results on existence and nonexistence of solutions.