| Title and Author(s) |
Page |
|
On curved squeezing and Conley index
Krzysztof P. Rybakowski
| ABSTRACT.
We consider reaction-diffusion equations on a family of domains depending
on a parameter $\eps>0$. As $\eps\to 0$, the domains degenerate to a lower
dimensional manifold. Using some abstract results introduced in the recent
paper \cite{\rfa{CR2}} we show that there is a limit equation as $\eps\to 0$
and obtain various convergence and admissibility results for the corresponding
semiflows. As a consequence, we also establish singular Conley index and
homology index continuation results.
Under an additional dissipativeness assumption, we also prove existence
and upper-semicontinuity of global attractors.
The results of this paper extend and refine
earlier results of \cite{\rfa{CR1}} and \cite{\rfa{PRR}}.
|
|
207
|
|
|
Constant-sign and nodal solutions for a Neumann problem with p-Laplacian and equi-diffusive reaction term
Salvatore A. Marano and Nikolaos S. Papageorgiou
| ABSTRACT.
The existence of both constant and sign-changing (namely, nodal)
solutions to a Neumann boundary-value problem with $p$-Laplacian
and reaction term depending on a positive parameter is
established. Proofs make use of sub- and super-solution techniques
as well as critical point theory.
|
|
233
|
|
|
On noncoercive periodic systems with vector p-Laplacian
Petru Jebelean and Nikolaos S. Papageorgiou
| ABSTRACT.
We consider nonlinear periodic systems driven by the vector
$p$-Laplacian. An existence and a multiplicity theorem are proved. In the
existence theorem the potential function is $p$-superlinear, but in general
does not satisfy the AR-condition. In the multiplicity theorem the
problem is strongly resonant with respect to the principal eigenvalue
$\lambda_0=0$. In both of the cases the Euler-Lagrange functional is
noncoercive and the method is variational.
|
|
249
|
|
|
Topological methods for boundary value problems involving discrete vector $\phi$-Laplacians
Cristian Bereanu and Dana Gheorghe
| ABSTRACT.
In this paper, using Brouwer degree arguments, we prove some
existence results for nonlinear problems of the type
-\nabla[\phi(\Delta x_m)]=g_m(x_m,\Delta x_m) \quad (1\leq m\leq n-1),
submitted to Dirichlet, Neumann or periodic boundary conditions,
where $\phi(x)=|x|^{p-2}x$ $(p>1)$ or $\phi(x)={x}/{\sqrt{1-|x|^2}}$
and $g_m\colon \RR^N\to\RR^N$ $(1\leq m\leq n-1)$ are continuous
nonlinearities satisfying some additional assumptions.
|
|
265
|
|
|
Homoclinic solutions of difference equations with variable exponents
Mihai Mihailescu, Vicentiu D. Radulescu and Stepan Tersian
| ABSTRACT.
We study the existence of homoclinic solutions for a class
of non-homogeneous difference equation
with periodic coefficients. Our proofs rely on the critical point
theory combined with adequate variational techniques, which are mainly
based on the mountain-pass lemma.
|
|
277
|
|
|
Bifurcation of Fredholm maps II; the index bundle and bifurcation
Jacobo Pejsachowicz
| ABSTRACT.
We obtain an estimate for the covering dimension of the set
of bifurcation points for solutions of nonlinear elliptic
boundary value problems from the principal symbol of the linearization
along the trivial branch of solutions.
|
|
291
|
|
|
Existence of multi-peak solutions for a class of quasilinear problems in R^N
Claudianor Oliveira Alves
| ABSTRACT.
Using variational methods we establish existence of multi-peak solutions
for the following class of quasilinear problems
-\varepsilon^{p}\Delta_{p}u + V(x)u^{p-1}= f(u), \quad u>0,
\text{ in } {\Bbb R}^{N}
where $\Delta_{p}u$ is the $p$-Laplacian operator, $2 \leq p < N$,
$\varepsilon >0$ and $f$ is a continuous function with
subcritical growth.
|
|
307
|
|
|
Optimal regularity of stable manifolds of nonuniformly hyperbolic dynamics
Luis Barreira and Claudia Valls
| ABSTRACT.
We establish the existence of smooth invariant stable manifolds
for differential equations $u'=A(t)u+f(t,u)$
obtained from sufficiently small perturbations of
a {\it nonuniform}\/ exponential dichotomy for the linear equation
$u'=A(t)u$. One of the main advantages of our work is that the results are
optimal, in the sense that the invariant manifolds are of
class $C^k$ if the vector field is of class $C^k$. To the best of
our knowledge, in the nonuniform setting this is the first general
optimal result (for a large family of perturbations and not for
some specific perturbations). Furthermore, in contrast to some
former works, we do not require a strong nonuniform exponential
behavior (we note that contrarily to what happens for autonomous
equations, in the nonautonomous case a nonuniform exponential
dichotomy need not be strong). The novelty of our proofs, in this
setting, is the use of the fiber contraction principle to
establish the smoothness of the invariant manifolds. In addition,
we can also consider linear perturbations, and our results have
thus immediate applications to the robustness of nonuniform
exponential dichotomies.
|
|
333
|
|
|
A one dimensional problem related to the symmetry of minimisers for the Sobolev trace constant in a ball
Olaf Torne
| ABSTRACT.
The symmetry of minimisers for the best constant in the trace inequality
in a ball,
$S_q(\rho)=\inf_{u\in W^{1,p}(B_\rho)} \|u\|^p_{W^{1,p}(B_\rho)}/
\|u\|^{p}_{L^q(\partial B(\rho))}$ has been studied by various authors.
Partial results are known which imply radial symmetry of minimisers, or
lack thereof, depending on the values of trace exponent $q$ and the radius
of the ball $\rho$. In this work we consider a one dimensional analogue
of the trace inequality and the corresponding minimisation problem for
the best constant. We describe the exact values of $q$ and $\rho$
for which minimisers are symmetric. We also consider the behaviour
of minimisers as the symmetry breaking threshold for $q$ and $\rho$
is breached, and show a case in which both symmetric and nonsymmetric
minimisers coexist.
|
|
363
|
|
|
Conley index of isolated equilibria
Martin Kell
| ABSTRACT.
In this paper we study stable isolated invariant sets and show that
the zeroth singular homology of the Conley index characterizes stability
completely. Furthermore, we investigate isolated mountain pass points
of gradient-like semiflows introduced by Hofer in \cite{4}
and show that the first singular homology characterizes them completely.
The result of the last section shows that for reaction-diffusion equations
u_{t}-\Delta u&\, = f(u),
u_{|\partial\Omega}&\, = 0,
the Conley index of isolated mountain pass points is equal to $\Sigma^{1}$
- the pointed $1$-sphere. Finally we generalize the result
of {\cite{1, Proposition 3.3}}
about mountain pass points to Alexander-Spanier cohomology.
|
|
373
|
|
|
Uniformly bounded composition operators between general Lipschitz function normed spaces
Janusz Matkowski
| ABSTRACT.
The notions of uniform boundedness and equidistant uniform boundedness of an
operator (both weaker then usual boundedness) are introduced. The main
results say that the generator of any uniformly bounded (or equidistantly
uniformly bounded) composition Nemytski{\u\i} operator acting between general
Lipschitzian normed function spaces must be affine with respect to the
function variable.
|
|
395
|
|
|
Location of fixed points in the presence of two cycles
Alfonso Ruiz-Herrera
| ABSTRACT.
Any orientation-preserving homeomorphism of the plane having a two
cycle has also a fixed point. This well known result does not
provide any hint on how to locate the fixed point, in principle it
can be anywhere. J. Campos and R. Ortega in {\it Location of fixed
points and periodic solutions in the plane} consider the class of
Lipschitz-continuous maps and locate a fixed point in the region
determined by the ellipse with foci at the two cycle and
eccentricity the inverse of the Lipschitz constant. It will be
shown that this region is not optimal and a sub-domain can be
removed from the interior. A curious fact is that the ellipse
mentioned above is relevant for the optimal location of fixed
point in a neighbourhood of the minor axis but it is of no
relevance around the major axis.
|
|
407
|
