Sergio Rolando

A nonlinear elliptic equation with singular potential and applications to nonlinear field equations

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.

A note on nonlinear elliptic problems with singular potentials

We deal with the semi-linear elliptic problem −4u + V (|x|)u = f (u) , u ∈ D(R ;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.

Compactness and existence results in weighted Sobolev spaces of radial functions. Part I: compactness

Given two measurable functions

A note on vortices with prescribed charge

V(r )\ge 0

Vortices with Prescribed L2 Norm in the Nonlinear Wave Equation

V(r)≥0 and

Sum of weighted Lebesgue spaces and nonlinear elliptic equations

K(r)> 0