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TOPOLOGICAL METHODS
IN
NONLINEAR ANALYSIS
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| Vol. 25, No. 1 March 2005 |
TABLE OF CONTENTS
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Title and Author(s) |
Page |
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On the Schrodinger equation involving a critical Sobolev exponent and magnetic field
Jan Chabrowski and Andrzej Szulkin
| ABSTRACT.
We consider the semilinear Schrodinger equation
-\Delta_A u + V(x)u = Q(x)|u|^{2^{*}-2}u.
Assuming that V changes sign, we establish the
existence of a solution u\ne 0 in the Sobolev space H_{A,V^+}^{1}(\RN).
The solution is obtained by a min-max type argument based on
a topological linking. We also establish certain regularity properties
of solutions for a rather general class of equations involving the
operator -\Delta_A.
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3
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Matter and electromagnetic fields: remarks on dualistic and unitarian standpoints
Vieri Benci and Donato Fortunato
| ABSTRACT.
The study of the relation of matter and the electromagnetic field is a
classical, intriguing problem both from physical and mathematical point of
view. This relation can be interpreted
from two different standpoints which, following [5], are called
unitarian standpoint and dualistic standpoint.
In this paper we briefly describe two models which are related to the
unitarian and the dualistic standpoint respectively. For each model it is
possible to prove the existence of solitary waves which can be interpreted as
matter particles.
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23
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Conditional energetic stability of gravity solitary waves in the presence of weak surface tension
Boris Buffoni
| ABSTRACT.
For a sequence of values of the total
horizontal impulse that converges to 0, there are solitary
waves that minimise the energy in a given neighbourhood
of the origin in W2,2(R).
The problem arises in the framework
of the classical Euler equation when a two-dimensional
layer of water above an infinite horizontal bottom is considered, at the surface of which
solitary waves propagate under the action of gravity and weak
surface tension.
The adjective "weak" refers to the Bond number, which is
assumed to be sub-critical (<1/3).
This extends previous results on the conditional energetic
stability of solitary waves
in the super-critical case, namely those by A. Mielke ([7]) and
by the author ([1]). Like in the latter, the method is based
on direct minimisation and concentrated compactness, but without
relying on "strict sub-additivity", which is still unsettled
in the present case. Instead, a complete and careful
analysis of minimising sequences
is performed that allows us to reach a conclusion, based only on the
non-existence of "vanishing" minimising sequences.
However, in contrast with [1],
we are unable to prove the existence
of minimisers for all small values of the total horizontal impulse.
In fact more is needed to get stability, namely that every minimising
sequence has a subsequence that converges to a global minimiser, after
possible shifts in the horizontal direction. This will be obtained
as a consequence of the analysis of minimising sequences.
Then exactly the same argument as in [1] gives conditional energetic
stability and is therefore
not repeated.
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41
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A Morse index theorem and bifurcation for perturbed geodesics on semi-Riemannian manifolds
Monica Musso, Jacobo Pejsachowicz and Alessandro Portaluri
| ABSTRACT.
Perturbed geodesics are trajectories of particles moving on
a semi-Riemannian manifold in the presence of a potential. Our
purpose here is to extend to perturbed geodesics on
semi-Riemannian manifolds the well known Morse Index Theorem. When
the metric is indefinite, the Morse index of the energy
functional becomes infinite and hence, in order to obtain a
meaningful statement, we substitute the Morse index by its
relative form, given by the spectral flow of an associated family
of index forms. We also introduce a new counting for conjugate
points, which need not to be isolated in this context, and prove
that our generalized Morse index equals the total number of
conjugate points. Finally we study the relation with the Maslov
index of the flow induced on the Lagrangian Grassmannian.
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69
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Boundary value problems for first order systems on the half-line
Patric J. Rabier and Charles A. Stuart
| ABSTRACT.
We prove existence theorems for first order boundary value problems on
(0,\infty), of the form \dot{u}+F(\,\cdot\,,u)=f, Pu(0)=\xi, where the
function F=F(t,u) has a t-independent limit F^{\infty}(u) at infinity
and P is a given projection. The right-hand side f is in L^{p}
((0,\infty),{\Bbb R}^{N}) and the solutions u are sought in
W^{1,p}((0,\infty),{\Bbb R}^{N}), so that they tend to 0 at infinity. By
using a degree for Fredholm mappings of index zero, we reduce the existence
question to finding a priori bounds for the solutions. Nevertheless,
when the right-hand side has exponential decay, our existence results are
valid even when the governing operator is not Fredholm.
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101
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Sharp Sobolev inequality involving a critical nonlinearity on a boundary
Jan Chabrowski and Jianfu Yang
| ABSTRACT.
We consider the solvability of the Neumann problem for the equation
-\Delta u + \lambda u = 0, \qquad
\frac {\partial u}{\partial \nu} = Q(x)|u|^{q - 2}u
on \partial \Omega, where Q is a positive and
continuous coefficient on \partial \Omega, \la is a parameter and
q = {2(N - 1)}/{(N - 2)} is a critical Sobolev exponent for the trace
embedding of H^1(\Omega) into L^q(\partial \Omega).
We investigate the joint effect of the mean curvature of \partial
\Omega and the shape of the graph of Q on the existence of solutions.
As a by product we establish a sharp Sobolev inequality for the trace
embedding. In Section 6 we establish the existence of solutions when
a parameter \lambda interferes with the spectrum of -\Delta with the
Neumann boundary conditions. We apply a min-max principle based on
the topological linking.
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135
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Multiplicity of positive solutions for semilinear elliptic problems with antipodal symmetry
Norimichi Hirano
| ABSTRACT.
In this paper, we show the multiple existence of positive solutions of
semilinear elliptic problems of the form
-\Delta u = \vert u\vert ^{2^{*}-2}u + f, \quad u\in H_{0}^{1}(\Omega),
where \Omega\subset{\Bbb R}^{N} is a bounded domain, 2^{*} is the
Sobolev critical exponent and f\in L^{2}(\Omega).
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155
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On trajectories of analytic gradient vector fields on analytic manifolds
Aleksandra Nowel and Zbigniew Szafraniec
| ABSTRACT.
Let f: M\to {\R} be an analytic proper function defined in
a neighbourhood of a closed "regular" (for instance semi-analytic or
sub-analytic) set P\subset f^{-1}(y).
We show that the set of non-trivial trajectories of the equation
\dot x = \nabla f(x) attracted by P has the same Cech-Alexander
cohomology groups as \Omega\cap\{f < y \},
where \Omega is an appropriately choosen neighbourhood of P. There are also
given necessary conditions for existence of a trajectory joining two
closed "regular" subsets of M.
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167
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Asymptotic bifurcation problems for quasilinear equations-existence and multiplicity results
Pavel Drabek
ABSTRACT.
In this paper we address the existence and multiplicity
results for
\cases
-\Delta_p u -\lambda |u|^{p-2} u = h (x,u) &\text{in }\Omega,
u = 0 &\text{on } \partial \Omega,
\endcases
where p>1, \Delta_p u = \text{\rm div}(|\nabla u|^{p-2}\nabla u),
h is a bounded function and the spectral parameter \lambda stays "near" the
principal eigenvalue of the p-Laplacian.
We show how the bifurcation theory combined with certain asymptotic
estimates yield desired results.
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183
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The Conley index and spectral sequences
Piotr Bartlomiejczyk
| ABSTRACT.
We define spectral sequences associated with Morse decompositions
of a compact metric space. We prove the existence and uniqueness
of such spectral sequences for continuous dynamical systems.
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195
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