
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 nonexistence 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. Apriori
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
BénardMarangoni 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 LotkaVolterra
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
onedimensional
(
'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



