# 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

## Abstract

We deal with the semi-linear elliptic problem

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

where the potential $V>0$ 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.

## Cite this article

Marino Badiale, Sergio Rolando, A note on nonlinear elliptic problems with singular potentials. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 17 (2006), no. 1, pp. 1–13

DOI 10.4171/RLM/450