JournalsrlmVol. 17 , No. 1DOI 10.4171/rlm/450

A note on nonlinear elliptic problems with singular potentials

  • Marino Badiale

    Università degli Studi di Torino, Italy
  • Sergio Rolando

    Università degli Studi di Torino, Italy
A note on nonlinear elliptic problems with singular potentials cover

Abstract

We deal with the semi-linear elliptic problem

u+V(x)u=f(u),uD1,2(RN;R)-\triangle u+V\left( \left| x\right| \right) u=f\left( u\right) ,\quad u\in D^{1,2}(\mathbb{R}^{N};\mathbb{R})

where the potential V>0V>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 ff is odd, we show that the problem has infinitely many radial solutions. Nonexistence results for particular potentials and nonlinearities are also given.