My reserch interests are focused in Nonlinear Partial Differential Equations and Bifurcation Theory

Subcritical nonlinearities for elliptic equations

In  collaboration with A. Castro, from Harvey Mudd College, Claremont, CA, USA, we work on the existence of positive solutions for the subcritical elliptic equation. It is well known that Pohozaev identity provides non-existence result for an star shaped domain and a critical exponent in the nonlinearity. Any convex set is star shaped, and  a ring is not. The exponent is critical for the compactness of Sobolev imbedding's. A-priori bounds, combined with topological degree plays a fundamental rol.

Localized boundary conditions

In  collaboration with A. L. Pereira (Universidade de Sao Paulo, Brazil) and J. Sabina (Universidad de La Laguna) we work on  the differentiability  of the principal eigenvalue to the localized boundary problem. The lack of regularity  up to the boundary of the first derivative of the principal eigenfunctions is a further intrinsic feature of the problem. The study is of interest in mathematical models in morphogenesis.

 Nonlinear boundary conditions

In  collaboration with J. M. Arrieta and A. Rodriguez - Bernal, from Universidad Complutense, we work on nonlinear boundary conditions  with sub - linear boundary terms and possibly oscillatory. In particular, we determine conditions for the existence of unbounded solutions, we characterize its stability, and the existence of turning points and of resonant solutions. Also in  collaboration with A. Castro, from Harvey Mudd College, Claremont, CA, USA, we work on oscillatory nonlinearities. We determine conditions for the existence of  turning points and infinites changes of stability, without having any resonant solutions.

Non linear Schrodinger equation

In collaboration  with Victor Pérez, from the ‘Universidad de Castilla la Mancha’, we study the nonlinear Schrodinger equations with  spatially inhomogeneous interactions. We show that theground state experiences a strong localization on the spatial region where the interactions vanish. We apply this to phenomena like Bose – Einstein condensates

The Benard - Marangoni problem

We are developing a nascent research's lines in collaboration with Henar Herrero (Universidad de Castilla la Mancha) and Sergio Hoyas (Universidad Politécnica de Valencia). The Bénard-Marangoni problem is a physical phenomenon of thermal convection in which the effects of buoyancy and surface tension are taken into account. This problem is modelled as a boundary problem: a system of partial differential equations including Navier Stokes's like equations and a heat's like equation witha transport term, and  boundary conditions including crossed boundary condition which involves tangential derivatives of the temperature and velocity fields.

Reaction - diffusion systems and Lotka - Volterra systems

In  collaboration with J. López Gómez (Universidad Complutense) we consider Lotka-Volterra  problem  with diffusion considering spatial and transport terms.  Also with S. Merino (Universitat Zurich), we discuss the existence and uniqueness states for the problem of coexistence of predator - prey. The existence results are obtained in a very general context. The uniqueness result, has a restriction in the number of spatial coordinates: the proof is only achieved in the one-dimensional ( 'rivers' and not seas). This restriction is apparently technical although, today, and within of my knowledge, has not been surpassed in the general case.

The p - laplacian

In collaboration work with J. Fleckinger and F. of Thélin (Université de Toulouse), I made a incursion in  nonlinear diffusion phenomena that do not follow Newton's laws, as some gels.

 
Publications