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.

A note on nonlinear elliptic problems with singular potentials

Marino Badiale

Sergio Rolando

Abstract

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