Maria J. Esteban

Existence and non-existence results for semilinear elliptic problems in unbounded domains

In this paper, we prove various existence and non-existence results for semilinear elliptic problems in unbounded domains. In particular we prove for general classes of unbounded domains that there exists no solution distinct from 0 of for any smooth f satisfying f (0) = 0. This result is obtained by the use of new identities that solutions of semilinear elliptic equations satisfy.

Stationary States of the Nonlinear Dirac Equation: A Variational Approach

In this paper we prove the existence of stationary solutions of some nonlinear Dirac equations. We do it by using a general variational technique. This enables us to consider nonlinearities which are not necessarily compatible with symmetry reductions.

Stationary Solutions of Nonlinear Schrödinger Equations with an External Magnetic Field

In this paper we study the existence of stationary solutions of some Schrodinger equations with an external magnetic field. We obtain these solutions by solving appropriate minimization problems for the corresponding energy-functional. These problems which are a priori not compact are solved by the use of the so-called concentration-compactness method. We also prove the existence of solutions for the generalized Hartree-Fock equations which model the interaction of electrons and static nucleii through a coulombic potential and under the action of an external magnetic field.

Stationary solutions of the Maxwell-Dirac and the Klein-Gordon-Dirac equations

The Maxwell-Dirac system describes the interaction of an electron with its own electromagnetic field. We prove the existence of soliton-like solutions of Maxwell-Dirac in (3+1)-Minkowski space-time. The solutions obtained are regular, stationary in time, and localized in space. They are found by a variational method, as critical points of an energy functional. This functional is strongly indefinite and presents a lack of compactness. We also find soliton-like solutions for the Klein-Gordon-Dirac system, arising in the Yukawa model.

Nonlinear eigenvalues and bifurcation problems for Pucci's operators

In this paper we extend existing results concerning generalized eigenvalues of Puccis extremal operators. In the radial case, we also give a complete description of their spectrum, together with an equivalent of Rabinowitzs Global Bifurcation Theorem. This allows us to solve nonlinear equations involving Puccis operators.

Variational methods in relativistic quantum mechanics

This review is devoted to the study of stationary solutions of lin- ear and nonlinear equations from relativistic quantum mechanics, involving the Dirac operator. The solutions are found as critical points of an energy func- tional. Contrary to the Laplacian appearing in the equations of nonrelativistic quantum mechanics, the Dirac operator has a negative continuous spectrum which is not bounded from below. This has two main consequences. First, the energy functional is strongly indefinite. Second, the Euler-Lagrange equations are linear or nonlinear eigenvalue problems with eigenvalues lying in a spectral gap (between the negative and positive continuous spectra). Moreover, since we work in the space domain R 3 , the Palais-Smale condition is not satisfied. For these reasons, the problems discussed in this review pose a challenge in the Calculus of Variations. The existence proofs involve sophisticated tools from nonlinear analysis and have required new variational methods which are now applied to other problems. In the first part, we consider the fixed eigenvalue problem for models of a free self-interacting relativistic particle. They allow to describe the localized state of a spin-1/2 particle (a fermion) which propagates without changing its shape. This includes the Soler models, and the Maxwell-Dirac or Klein- Gordon-Dirac equations. The second part is devoted to the presentation of min-max principles al- lowing to characterize and compute the eigenvalues of linear Dirac operators with an external potential, in the gap of their essential spectrum. Many con- sequences of these min-max characterizations are presented, among them a new kind of Hardy-like inequalities and a stable algorithm to compute the eigenvalues. In the third part we look for normalized solutions of nonlinear eigenvalue problems. The eigenvalues are Lagrange multipliers, lying in a spectral gap. We review the results that have been obtained on the Dirac-Fock model which is a nonlinear theory describing the behavior of N interacting electrons in an external electrostatic field. In particular we focus on the problematic definition of the ground state and its nonrelativistic limit. In the last part, we present a more involved relativistic model from Quan- tum Electrodynamics in which the behavior of the vacuum is taken into ac- count, it being coupled to the real particles. The main interesting feature of this model is that the energy functional is now bounded from below, providing us with a good definition of a ground state.

A direct variational approach to Skyrme's model for meson fields

The solutions of Skyrmes variational problem describe the structure of mesons in a field of weak energy. The problem consists in minimizing the corresponding energy among the functions from ℝ3 toS3 which have a fixed “degree” without making any symmetry assumptions. We prove the existence of minima and study their properties.

On the modified Enskog equation for elastic and inelastic collisions. Models with spin

Abstract Under appropriate assumptions on the collision kernel we prove the existence of global solutions of the Enskog equation with elastic or inelastic collisions. We consider also this equation with spin, that is, the case when the angular velocities of the colliding particles are taken into account. In this case we also prove global existence results.

Multiple solutions of semilinear elliptic problems in a ball

In this paper lower bounds for the number of solutions of semilinear elliptic problems in a ball of RN are given. Its hypotheses are only related to the behavior of the nonlinearities at ±∞ and at 0. Global assumptions are never made. For example, oddness is never required for the proof of multiplicity results.

ON THE SYMMETRY OF EXTREMALS FOR THE CAFFARELLI-KOHN-NIRENBERG INEQUALITIES

Abstract In this paper we prove some new symmetry results for the extremals of the Caffarelli-Kohn-Nirenberg inequalities, in any dimension larger or equal to 2.