Andrea Sfecci

Periodic Bouncing Solutions for Nonlinear Impact Oscillators

Abstract We prove the existence of a periodic solution to a nonlinear impact oscillator, whose restoring force has an asymptotically linear behavior. To this aim, after regularizing the problem, we use phase-plane analysis, and apply the Poincaré-Bohl fixed point Theorem to the associated Poincaré map, so to find a periodic solution of the regularized problem. Passing to the limit, we eventually find the “bouncing solution” we are looking for.

On a diffusion model with absorption and production

Abstract We discuss the structure of radial solutions of some superlinear elliptic equations which model diffusion phenomena when both absorption and production are present. We focus our attention on solutions defined in R (regular) or in R ∖ { 0 } (singular) which are infinitesimal at infinity, discussing also their asymptotic behavior. The phenomena we find are present only if absorption and production coexist, i.e., if the reaction term changes sign. Our results are then generalized to include the case where Hardy potentials are considered.

Entire Solutions of Superlinear Problems with Indefinite Weights and Hardy Potentials

We provide the structure of regular/singular fast/slow decay radially symmetric solutions for a class of superlinear elliptic equations with an indefinite weight. In particular we are interested in the case where such a weight is positive in a ball and negative outside, or in the reversed situation. We extend the approach to elliptic equations in presence of Hardy potentials, i.e. to

Periodic solutions of weakly coupled superlinear systems

\begin{aligned} \varDelta u +\frac{h(|\text {x}|)}{|\text {x}|^2} u+ f(u, |\text {x}|)=0 \end{aligned}

On a singular periodic Ambrosetti–Prodi problem

Δu+h(|x|)|x|2u+f(u,|x|)=0where h is not necessarily constant. By the use of Fowler transformation we study the corresponding dynamical systems, presenting the construction of invariant manifolds when the global existence of solutions is not ensured.

Double resonance for one-sided superlinear or singular nonlinearities

Abstract We investigate the existence of periodic trajectories of a particle, subject to a central force, which can hit a sphere or a cylinder. We will also provide a Landesman–Lazer-type condition in the case of a nonlinearity satisfying a double resonance condition. Afterwards, we will show how such a result can be adapted to obtain a new result for the impact oscillator at double resonance.