I discuss the rigorous derivation of the time-dependent magnetic Hartree equation that describes the mean-field dynamics of rotating bosons in confining traps.

Let G denote the image of a smooth embedding of the circle S¹ in R^N, N ³ 2. Denote by W_R the open normal tubular neighborhood of RG of radius 1. Consider the superlinear problem -Du = f(u) on the expanding domains W_R (i.e., as R®¥) with homogeneous Dirichlet boundary conditions. We prove the existence of multibump solutions with bumps lined up along RG and with alternating signs. Here we allow nonodd functions f.

The two dimensional two-well problem arises in the study of the zero energy states of a solid-solid phase transition in materials that exhibit the so-called shape memory effect.

This problem can be formulated as follows: find u : W® R³ such that

For a suitable energy and in the proper regime, we show that the nonzero energy states for the linearized version of (1), are also closed to a simple laminate.

We consider a class of Caffarelli-Kohn-Nirenberg inequalities without restricting the pertinent parameters and determine the values of the optimal constants and the functions that achieve them, i.e., minimizers of a suitable functional. By studying a corresponding Euler-Lagrange equation, we also find infinitely many sign-changing solutions at higher energy levels in addition to the ground-state solutions.

We examine the equation

This equation is of practical interest since it is the steady state of an equation modeling a simple Micro-Electro-Mechanical System (MEMS) device.

[no abstract supplied]

We investigate the bifurcation of periodic solutions from relative equilibria, examples being the n body problem or the n vortex problem. We use the approach of orthogonal degree theory, which lets us probe the existence of global symmetric branches of periodic solutions. We particularly report a general result of bifurcation on the equation of a satellite influenced by a relative equilibria of primaries. We will discuss further the case in which the primaries form a 1+n-gon, like the Maxwell model for the Saturn rings. We also discuss the case of Halo orbits in the restricted tree body problem.

The concentration of calcium ions inside dendritic spines (microstructures of the neuron) plays a crucial role in the synaptic plasticity, and in consequence in cognitive processes like learning and memory. We construct a reaction-diffusion system that models the dynamics of calcium ions in the spine, taking into account the chemical interactions between the calcium ions and three different types of proteins. We prove that this system is a well-posed problem, i.e., we have a priori estimates, global existence, global uniqueness, positivity of solutions and continuity with respect of the initial data.

This result will appear in the article of Kamel Hamdache and Mauricio Labadie, On a reaction-diffusion model for calcium dynamics in dendritic spines, Nonlinear Analysis: Real World Applications 10(2009), 2478-2492 (August issue). This article has been published online on May 2008 (doi: 10.1016/j.nonrwa.2008.05.005).

The existence of at least two solutions for a resonance problem involving the p-Laplacian is shown for the case of bounded domains in R^N. This work constitutes an extension of a previous result of Landesman, Robinson and Rumbos for the case p=2 (Nonlinear Analysis TMA 24(1995), 1049-1059).

The inhomogeneous Robin/third boundary condition with general coefficient for the Poisson equation on the unit disc is studied in terms of holomorphic functions using Fourier analysis. It is shown that against the usual expectations this problem cannot have a unique solution unless the coefficient of the first order term in the boundary condition is a constant. For the case of general coefficient, it is actually a problem with essential singularity in the domain, but still well-posed under proper assumptions and the unique solution is given explicitly.

We give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball B of radius R in Rⁿ, n ³ 1, so that the following inequalities hold for all u Î C₀^¥(B):

We present some preliminary results on the continuation and bifurcations of breathers in the discrete NLS in a finite one-dimensional lattice. We show numerical evidence for fold and pitchfork bifurcations as the intesite coupling increases and also discuss breathers that can be continued to normal modes of the weakly linear system.

In this talk we briefly present recent results regarding the existence of sign-changing solutions of the semilinear elliptic problem

The most of these developments are a joint work with Alfonso Castro and Jorge Cossio.