Enrico Serra

Semilinear Elliptic Equations for Beginners

Introduction and basic results.- Minimization techniques: compact problems.- Minimization techniques: lack of compactness.- Introduction to minimax methods.- Index of the main assumptions

Multiplicity results for the supercritical Henon equation

Abstract We study the Dirichlet problem for the Hénon equation ¡Δu = |x|αup-1 in the unit ball B ⊂ RN. For N ≥ 4 and α large we prove the existence of positive nonradial solutions for a range of ps including supercritical values.

Collisionless periodic solutions to some three-body problems

We provide sufficient conditions for the existence of periodic solutions to some three-body problems. Periodic solutions are found as minima of the associated action integral and are shown to be free of double and triple collisions.

Non-continuation of the periodic oscillations of a forced pendulum in the presence of friction

A well known theorem says that the forced pendulum equation has periodic solutions if there is no friction and the external force has mean value zero. In this paper we show that this result cannot be extended to the case of linear friction.

On the existence of homoclinic solutions for almost periodic second order systems

Abstract In this paper we prove the existence of at least one homoclinic solution for a second order Lagrangian system, where the potential is an almost periodic function of time. This result generalizes existence theorems known to hold when the dependence on time of the potential is periodic. The method is of a variational nature, solutions being found as critical points of a suitable functional. The absence of a group of symmetries for which the functional is invariant (as in the case of periodic potentials) is replaced by the study of problems “at infinity” and a suitable use of a property introduced by E. Sere.

A note on the radial solutions for the supercritical Hénon equation

We prove the existence of a positive radial solution for the Henon equation with arbitrary growth. The solution is found by means of a shooting method and turns out to be an increasing function of the radial variable. Some numerical experiments suggest the existence of many positive oscillating solutions.

A variational approach to chaotic dynamics in periodically forced nonlinear oscillators

Abstract We prove that a class of problems containing the classical periodically forced pendulum equation displays the main features of chaotic dynamics. The approach is based on the construction of multibump type heteroclinic solutions to periodic orbits by the use of global variational methods.

Negative Energy Ground States for the L2-Critical NLSE on Metric Graphs

We investigate the existence of ground states with prescribed mass for the focusing nonlinear Schrödinger equation with L2-critical power nonlinearity on noncompact quantum graphs. We prove that, unlike the case of the real line, for certain classes of graphs there exist ground states with negative energy for a whole interval of masses. A key role is played by a thorough analysis of Gagliardo–Nirenberg inequalities and by estimates of the optimal constants. Most of the techniques are new and suited to the investigation of variational problems on metric graphs.

Symmetry breaking results for problems with exponential growth in the unit disk

We investigate some asymptotic properties of extrema uα to the two- dimensional variational problem sup u∈H1 0(B) k uk =1 Z B � e γu 2 1 � |x| α dx as � ! +1. Here B is the unit disk of R 2 and 0 < � 4� is a given parameter. We prove that in a certain range of s, the maximizers are not radial forlarge.

Bound states of the NLS equation on metric graphs with localized nonlinearities

Abstract We investigate the existence of multiple bound states of prescribed mass for the nonlinear Schrodinger equation on a noncompact metric graph. The main feature is that the nonlinearity is localized only in a compact part of the graph. Our main result states that for every integer k , the equation possesses at least k solutions of prescribed mass, provided that the mass is large enough. These solutions arise as constrained critical points of the NLS energy functional. Estimates for the energy of the solutions are also established.