Vittorio Coti Zelati

Solutions with minimal period for Hamiltonian systems in a potential well.

Abstract Let U ∈ C2(Ω), where Ω is a bounded set in ℝN Suppose that U(x) tends to + ∞ as x tends to ∂Ω. Our main results concern the existence of periodic solutions of − x ¨ + U ′ ( x ) having a prescribed number T as minimal period. The results are also generalized to first order Hamiltonian systems.

Periodic Solutions of Singular Dynamical Systems

The paper contains a discussion on some recent advances in the existence of periodic solutions of some second order dynamical systems with singular potentials.

Symmetry breaking and multiple solutions for a Neumann problem in an exterior domain

In this paper we prove existence of multiple positive solutions for a Neumann problem in ℝ N / (0, R ), R large, with a superquadratic and odd nonlinearity. The proof is based on the fact that in such a situation the minimum of the corresponding energy functional (which is achieved) is not an even function and that there is quite a large gap (for large R ) between such a minimum and the minimum of the same functional on even functions. In the set of functions whose energy lies in such a gap, we can apply index theory to prove the desired multiplicity result.

Multiple brake orbits for some classes of singular Hamiltonian systems

from which one can easily construct a periodic solution of period 2T by reflecting q around t = 0, T. If Q = (x E E 1 V(x) < hJ is homeomorphic to a ball then the existence of one brake orbit is a classical result of Seifert [ 11. Generalizations of that result, as well as alternative proofs were given in [2-6, 141. In the case in which the topology of Sz is richer, then the existence of multiple solutions has been proved by Bolotin and Kozlov in [7] (see also [8]). In this paper we are mainly concerned with the case in which the potential V has singularities inside &2 (in the sense that V(x) + ---CO as x + x,, E ai) and the topology of Q is nontrivial. More precisely, in Section 2 we shall assume that Sz has a certain number of “holes” k and that V behaves like -lxI-O1 with 0 < cx < 2 near x = 0 and we shall show that (1.3) has at least k classical brake orbits. Let us point out that the main interest of this result lies in the fact that we are able to prove that the solutions we find are classical ones, that is they do not collide with the singularity, while most of the papers dealing with existence of fixed energy [9] or fixed period [lo] solutions for singular potentials only prove existence of generalized solutions, that is solutions which can collide with the singularity. Existence of a noncollision periodic solution for a singular potential

A Variational Approach to the Eigenfunctions of the One Particle Relativistic Hamiltonian

In this note we give a variational characterization of the eigenvalues and eigenvectors for the operator

Multiple homoclinic orbits for a class of conservative systems

H = {H_0} + V = \sqrt { - {c^2}\Delta + {m^2}{c^4}} + V,

Existence of ground states for nonlinear, pseudo-relativistic Schrödinger equations

H=H0+V=−c2Δ+m2c4+V, where H0) is the relativistic (free) Hamiltonian operator and V is a real valued potential. Our results hold when