Abbas Bahri

On the existence of a positive solution of semilinear elliptic equations in unbounded domains

We prove here the existence of a positive solution, under general conditions, for semilinear elliptic equations in unbounded domains with a variational structure. The solutions we build cannot be obtained in general by minimization problems. And even if Palais-Smale condition is violated, precise estimates on the losses of compactness are obtained by the concentration-compactness method which enables us to apply the theory of critical points at infinity.

Periodic solutions of Hamiltonian systems of 3-body type

Abstract We study the question of the existence of periodic solutions of Hamiltonian systems of the form: (✶) q ¨ + V q ( t , q ) = 0 where V = ∑ 1 ≦ i ≠ j 3 V i j ( t , q i − q j ) with V(t, ξ) T-periodic in t and singular at ξ = 0. Under hypotheses on V of 3-body type, we prove that the functional corresponding to (✶) has an unbounded sequence of critical points provided that the singularity of V at 0 is strong enough.

Periodic orbits in magnetic fields and Ricci curvature of Lagrangian systems

where dA -F, on the space of closed curves on the manifold Mn. Here A is a 1-form (i.e., F is an exact 2-form). This functional is a natural generalization of the usual functional of length, and its closed extremals correspond to periodic trajectories of the motion of particles on the Riemannian manifold Mn when the kinetic energy is defined by the metric tensor and the form F defines a magnetic field. Also this functional corresponds to the periodic orbits for other problems of classical mechanics and mathematical physics, as it was shown in [N2], [N3], [NS]. When the Lagrangian function

The difference of topology at infinity in changing-sign Yamabe problems on ?3 (the case of two masses)

In this paper, we study the simplest cases of differences of topology at infinity in Yamabe-type problems with changing-sign solutions. In the past twenty years, there has been a wide range of activity in the study of the positive solutions to the problems of Yamabe-type, using various methods, including the study of differences of topology at infinity. After [1, 6], these differences of topology are starting to be well understood in the framework of positive functions. In sharp contrast to this, very little is known when the hypothesis of positivity is removed. We believe that completing such a task is important not only per se, but also because it lays the ground for Yang-Mills (Einstein’s?) equations. These equations should only represent a complication in the background framework with respect to Yamabe changing-sign problems. In the present work, we are completing this program for the pure Yamabe problem—allowing for sign changes—on S3 and for only pairs of functions at infinity. In order to formulate our results, we need to introduce some notation and quote slight variations of well-known results. The partial differential equation that we will be studying is

A remark on the bifurcation diagrams of superlinear elliptic equations

We prove a formula relating the index of a solution and the rotation number of a certain complex vector along bifurcation diagrams.

Periodic Solutions of Some Problems of 3-Body Type

The study of time periodic solutions of the n-body problem is a classical one. See e.g 1 The purpose of this paper is to sketch some of our recent research on the existence of time periodic solutions of Hamiltonian systems of 3-body type [2] This work presents a new direct variational approach to the problem.

Removing Condition (A6)

We prove here that if x ∞is a critical point at infinity of index io + i ∞larger than or equal to 3, then the sequence of Morse indices of its iterates is strictly increasing, allowing us to focus on exactly one iterate in order to fulfill the conditions which warrant that our homology is well defined. Namely, letx∞ k be the iterate of x∞of order k. The Ho1-index ofx∞ k is clearlykio We denotei∞ k the index at infinity ofx∞kWe then have the following result.

Completion of the Removal of (A5)

We now prove that we can modify the contact formain the vicinity of a false critical point at infinity of the third kind \({\bar x^\infty }\), using the procedures introduced in the proofs of Proposition 16 and Lemma 11 so that such a critical point at infinity does not interfere with our homology.

An Outline for the Removal of (A2)

We now discuss condition (A2) and outline how this hypothesis can be removed and replaced by the much weaker (A2)’. Some computations are needed in order to carry out this removal and replacement, which we will not complete here. Although our arguments can be worked out, after more technical details, into a formal proof, we have kept one step away from this goal and part of our results will be presented under the form of claims, with sketches of proofs.

Introduction, Statement of Results, and Discussion of Related Hypotheses

This monograph is related to two previous ones, [1] and [2], in the same direction and is an attempt to create a new tool for the study of one or several aspects of the dynamics of a contact structure and a contact vector field in the family which it defines.