We study existence and asymptotic properties of solutions to a semilinear elliptic equation in the whole space. The equation has a cylindrical symmetry and we find cylindrical solutions. The main features of the problem are that the potential has a large set of singularities (i.e. a subspace), and that the nonlinearity has a double power-like behaviour, subcritical at infinity and supercritical near the origin. We also show that our results imply the existence of solitary waves with nonvanishing angular momentum for nonlinear evolution equations of Schrodinger and Klein-Gordon type.