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TOPOLOGICAL METHODS
IN
NONLINEAR ANALYSIS
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| Vol. 27, No. 1 March 2006 |
TABLE OF CONTENTS
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Title and Author(s) |
Page |
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On a second order boundary value problem with singular nonlinearity
Vieri Benci, Anna Maria Micheletti and Edlira Shteto
| ABSTRACT.
In this paper we investigate in a variational setting, the elliptic
boundary value problem $-\Delta u={\sign\,u}/{|u|^{\alpha+1}}$ in
$\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is an open connected
bounded subset of $\Real^N$, and $\alpha>0$. For the positive solution,
which is checked as a minimum point of the formally associated
functional
E(u)=\frac 12\int_\Omega|\nabla u|^2+\frac{1}{\alpha}
\int_\Omega \frac1{|u|^\alpha},
we prove dependence
on the domain $\Omega$. Moreover, an approximative functional $E_\eps$ is
introduced, and an upper bound for the sequence of mountain pass points
$u_\eps$ of $E_\eps$, as $\eps\to0$, is given. For the onedimensional
case, all sign-changing solutions of
$-u''={\sign\,u}/{|u|^{\alpha+1}}$ are characterized by their
nodal set as the mountain pass point and $n$-saddle points ($n>1$) of
the functional $E$.
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1
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Symmetric systems of Van Der Pol equations
Zelman Balanov, Meymanat Farzamirad and Wiesław Krawcewicz
| ABSTRACT.
We study the impact of
symmetries on the occurrence of periodic solutions in systems of
van der Pol equations. We apply the equivariant
degree theory to establish existence results for multiple nonconstant
periodic solutions and classify their symmetries. The computations of the
algebraic invariants in the case of dihedral, tetrahedral, octahedral and
icosahedral symmetries for a van der Pol system of equations are included.
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29
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Eigenvalue criteria for existence of multiple positive solutions of nonlinear boundary value problems of local and nonlocal type
Jeffrey R. L. Webb and K. Q. Lan
| ABSTRACT.
New criteria are established for the existence of multiple positive
solutions of a Hammerstein integral equation of the form
u(t)= \int_{0}^1 k(t,s)g(s)f(s,u(s))\,ds \equiv Au(t)
where $k$ can have discontinuities in its second variable and $g \in
L^{1}$.
These criteria are determined by the relationship between the
behaviour of $f(t,u)/u$ as $u$ tends to $0^+$ or $\infty$ and the
principal (positive) eigenvalue of the linear Hammerstein integral
operator
Lu(t)=\int_{0}^1 k(t,s)g(s)u(s)\,ds.
We obtain new results
on the existence of multiple positive solutions of a second order
differential equation of the form
u''(t)+g(t)f(t,u(t))=0 \quad\text{a.e\. on } [0,1],
subject to general separated boundary conditions and also to nonlocal
$m$-point boundary conditions. Our results are optimal in some cases.
This work contains several new ideas, and gives a {\it unified}
approach applicable to many BVPs.
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91
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Radial solutions of semilinear elliptic equations with broken symmetry
Anna Maria Candela, Giuliana Palmieri and Addolorata Salvatore
| ABSTRACT.
The aim of this paper is to prove the existence of infinitely many radial solutions of
a superlinear elliptic problem with rotational symmetry
and non-homogeneous boundary data.
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117
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The 8$\pi$-problem for radially symmetric solutions of a chemotaxis model in a disc
Piotr Biler, Grzegorz Karch, Philippe Laurencot and Tadeusz Nadzieja
| ABSTRACT.
We study the properties and the large time asymptotics of radially
symmetric solutions of a chemotaxis system in a disc of $\RR^2$ when the
parameter is either critical and equal to $8\pi$ or subcritical.
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133
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An eigenvalue semiclassical problem for the Schrodinger operator with an electrostatic field
Teresa D'Aprile
| ABSTRACT.
We consider the following system of Schr\"odinger--Maxwell
equations in the unit ball $B_1$ of $\R^3$
-\frac{\hbar^2}{2m}\Delta v+ e\phi v=\omega v,\quad
-\Delta\phi=4\pi e v^2
with the boundary conditions $ u=0$, $
\phi=g$ on $\partial B_1$, where $\hbar$, $m$, $e$,
$\omega>0$, $v$, $\phi\colon B_1\rightarrow \R$, $g\colon \partial B_1\to \R$.
Such system describes the interaction of a particle constrained to
move in $B_1$ with its own electrostatic field. We exhibit a family
of positive solutions $(v_\hbar, \phi_\hbar)$ corresponding
to eigenvalues $\omega_\hbar$ such that $v_\hbar$ concentrates
around some points of the boundary $\partial B_1$ which are
minima for $g$ when $\hbar\rightarrow 0$.
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149
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Positive solutions for a nonconvex elliptic Dirichlet problem with superlinear response
Andrzej Nowakowski and Aleksandra Orpel
| ABSTRACT.
The existence of bounded solutions of the Dirichlet problem for a ceratin
class of elliptic partial differential equations is discussed here.
We use variational methods based on the subdifferential theory and the
comparison principle for difergence form operators.
We present duality and variational principles for this problem.
As a consequences of the duality we obtain also the variational principle for
minimizing sequences of $J$ which gives a measure of a duality gap between primal and dual
functional for approximate solutions.
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177
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