A note on nonlinear elliptic problems with singular potentials

Abstract

We deal with the semi-linear elliptic problem

where the potential $V>0$ is measurable, singular at the origin and may also have a continuous set of singularities. The nonlinearity is continuous and has a super-linear power-like behaviour; both sub-critical and super-critical cases are considered. We prove the existence of positive radial solutions. If $f$ is odd, we show that the problem has infinitely many radial solutions. Nonexistence results for particular potentials and nonlinearities are also given.