Maurizio Garrione

A Landesman-Lazer-type condition for asymptotically linear second-order equations with a singularity

We consider the T -periodic problem { x′′ + g(t, x) = 0 x(0) = x(T ), x′(0) = x′(T ), where g : [0, T ]× ]0,+∞[→ R exhibits a singularity of a repulsive type at the origin, and an asymptotically linear behavior at infinity. In particular, for x large, g(t, x) is controlled from both sides by two consecutive asymptotes of the T -periodic Dancer-Fucik spectrum, with possible equality on one side. By means of a suitable Landesman-Lazer type condition, we prove the existence of a solution.

Periodic perturbations of Hamiltonian systems

Abstract We prove existence and multiplicity results for periodic solutions of Hamiltonian systems, by the use of a higher dimensional version of the Poincaré–Birkhoff fixed point theorem. The first part of the paper deals with periodic perturbations of a completely integrable system, while in the second part we focus on some suitable global conditions, so to deal with weakly coupled systems.

Nonlinear Resonance: a Comparison Between Landesman-Lazer and Ahmad-Lazer-Paul Conditions

Abstract We show that the Ahmad-Lazer-Paul condition for resonant problems is more general than the Landesman-Lazer one, discussing some relations with other existence conditions, as well. As a consequence, such a relation holds, for example, when considering resonant boundary value problems associated with linear elliptic operators, the p-Laplacian and, in the scalar case, with an asymmetric oscillator.

Structure Results for Semilinear Elliptic Equations with Hardy Potentials

Abstract We prove structure results for the radial solutions of the semilinear problem Δ ⁢ u + λ ⁢ ( | x | ) | x | 2 ⁢ u + f ⁢ ( u ⁢ ( x ) , | x | ) = 0 , \Delta u+\frac{\lambda(|x|)}{|x|^{2}}u+f(u(x),|x|)=0, where λ is a function and f is superlinear in the u-variable. As particular cases, we are able to deal with Matukuma potentials and with nonlinearities f having different polynomial behaviors at zero and at infinity. We give the complete picture for the subcritical, critical and supercritical cases. The technique relies on the Fowler transformation, allowing to deal with a dynamical system in ℝ 3 {{\mathbb{R}}^{3}} , for which elementary invariant manifold theory allows to draw the conclusions involving regular/singular and fast/slow-decay solutions.

Resonant Sturm–Liouville Boundary Value Problems for Differential Systems in the Plane

We study the Sturm-Liouville boundary value problem associated with the planar differential system Jz′ = ∇V (z) + R(t, z), where V (z) is positive and positively 2-homogeneous and R(t, z) is bounded. Assuming Landesman-Lazer type conditions, we obtain the existence of a solution in the resonant case. The proofs are performed via a shooting argument. Some applications to boundary value problems associated with scalar second order asymmetric equations are discussed. MSC 2010 Classification 34B15.