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TOPOLOGICAL METHODS
IN
NONLINEAR ANALYSIS
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| Vol. 34, No. 1 September 2009 |
TABLE OF CONTENTS
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Title and Author(s) |
Page |
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On a variant of the maximum principle involving radial p-Laplacian with applications to nonlinear eigenvalue problems and nonexistence results
Tomasz Adamowicz and Agnieszka Kałamajska
| ABSTRACT.
We obtain the variant of maximum principle for
radial solutions of $p$-harmonic equation
$-a\Delta_p(w)=\phi(w)$.
As a consequence of this result we prove
monotonicity of constant sign solutions, analyze the support
of the solutions and study their oscillations. The results are applied
to various type nonlinear eigenvalue problems and nonexistence theorems.
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1
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A priori bounds via the relative Morse index of solutions of an elliptic system
Miguel Ramos
| ABSTRACT.
We prove a Liouville-type theorem for entire solutions of the elliptic
system $-\Delta u = |v|^{q-2}v$, $-\Delta v=|u|^{p-2}u$ having finite
relative Morse index in the sense of Abbondandolo. Here, $p,q >2$ and
$1/p+1/q>(N-2)/N$. In particular, this yields a result on a priori bounds
in $L^{\infty}\times L^{\infty}$ for solutions of superlinear elliptic
systems obtained by means of min-max theorems, for both Dirichlet and
Neumann boundary conditions.
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21
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Multiple nontrivial solutions of Neumann p-Laplacian systems
Dumitru Motreanu and Kanishka Perera
| ABSTRACT.
We obtain multiple nontrivial solutions of Neumann $p$-Laplacian systems via Morse theory.
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41
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Multiple solutions for operator equations involving duality mappings on Orlicz-Sobolev spaces
George Dinca and Pavel Matei
| ABSTRACT.
Let $X$ be a real reflexive and separable Banach space having the
Kade\v{c}-Klee property, compactly imbedded in the real Banach space $V$ and
let $G\colon V\rightarrow {\Bbb R} $
be a differentiable functional.
By using ``fountain theorem'' and ``dual fountain theorem'' (Bartsch \cite{3}
and Bartsch-Willem \cite{4}, respectively), we will study the multiplicity
of solutions for operator equation
$$
J_{\varphi}u=G^{\prime}(u),
$$
where $J_{\varphi}$ is the duality mapping on $X$, corresponding to the gauge
function $\varphi$.
Equations having the above form with $J_{\varphi}$ a duality mapping on
Orlicz-Sobolev spaces are considered as applications. As particular cases of
the latter results, some multiplicity results concerning duality mappings on
Sobolev spaces are derived.
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49
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Multiplicity results for some quasilinear elliptic problems
Francisco Odair de Paiva, João Marcos do Ó and Everaldo Souto de Medeiros
| ABSTRACT.
In this paper, we study multiplicity of weak solutions for the
following class of quasilinear elliptic problems of the form
$$
-\Delta_p u -\Delta u = g(u)-\lambda |u|^{q-2}u
\quad \text{in } \Omega \text{ with } u=0 \text{ on } \partial\Omega,
$$
where $ \Omega $ is a bounded domain in ${\Bbb R}^n $ with
smooth boundary $\partial\Omega$, $ 1
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77
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Comparison results and existence of bounded solutions to strongly nonlinear second order differential equations
Cristina Marcelli and Francesca Papalini
| ABSTRACT.
We investigate the existence of bounded solutions on the whole real
line of the following strongly non-linear non-autonomous
differential equation
$$
(a(x(t))x'(t))'= f(t,x(t),x'(t)) \quad \text{a.e. } t\in \erre
\tag \text{\rm E}
$$
where $a(x)$ is a
generic continuous positive function, $f$ is a Carathe\'odory
right-hand side.
We get existence results by combining the upper and lower-solutions
method to fixed-point techniques. We also provide operative
comparison criteria ensuring the well-ordering of pairs of upper and
lower-solutions.
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91
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On a $p$-superlinear Neumann $p$-Laplacian equation
Sergiu Aizicovici, Nikolaos S. Papageorgiou and Vasile Staicu
| ABSTRACT.
We consider a nonlinear Neumann problem, driven by the $p$-Laplacian, and
with a nonlinearity which exhibits a $p$-superlinear growth near infinity,
but does not necessarily satisfy the Ambrosetti-Rabinowitz condition. Using
variational methods based on critical point theory, together with suitable
truncation techniques and Morse theory, we show that the problem has at least
three nontrivial solutions, of which two have a fixed sign (one positive and
the other negative).
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111
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Bounded solutions to nonlinear delay differential equations of third order
Cemil Tunç
| ABSTRACT.
This paper gives some sufficient conditions for every
solution of delay differential equation
$$
\multline
\buildrel \ldots \over x (t)
+f(t,x(t),x(t-r),\dot{x}(t),\dot{x}(t-r),\ddot{x}(t),\ddot{x}(t-r))
\\
+b(t)g(x(t-r),\dot{x}(t-r)) +c(t)h(x(t))
\\
=p(t,x(t),x(t-r),\dot{x}(t),\dot{x}(t-r),\ddot{x}(t))
\endmultline
$$
to be bounded.
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131
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Extension theorems and topological essentiality in $\alpha$-weakly convex metric spaces
F. S. de Blasi, Lech Górniewicz and G. Pianigiani
| ABSTRACT.
Paper is dedicated to Andrzej Granas on the occasion of his 80th birthday
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141
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Natural topologies on Colombeau algebras
J. Aragona, R. Fernandez and S. O. Juriaans
| ABSTRACT.
We define intrinsic, natural and metrizable topologies
${\Cal T}_{\Omega}$, ${\Cal T}$, ${\Cal T}_{s,\Omega}$
and ${\Cal T}_s$ in ${\Cal G}(\Omega)$, $\OK$, ${\Cal G}_s(\Omega)$
and $\OK_s$, respectively. The topology ${\Cal T}_{\Omega}$ induces
${\Cal T}$, ${\Cal T}_{s,\Omega}$ and ${\Cal T}_s$.
The topologies ${\Cal T}_{s,\Omega}$ and ${\Cal T}_s$
coincide with the Scarpalezos sharp topologies.
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161
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Uniform nonsquareness of direct sums of Banach spaces
Anna Betiuk-Pilarska and Stanisław Prus
| ABSTRACT.
An inequality between James constants of Banach spaces $X_s$
and the James constant
of their direct sum is obtained. This gives a characterization
of uniform nonsquareness of sums of Banach spaces.
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181
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Spectral Sequences and Detailed Connection Matrices
Piotr Bartłomiejczyk
| ABSTRACT.
We introduce detailed connection matrices.
We prove that the spectral sequence can be reconstructed
from a detailed connection matrix in the category of filtered
differential vector spaces.
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187
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