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Project Euclid (Hosted at Cornell University Library) (198.174 recursos)
Topological Methods in Nonlinear Analysis
Topological Methods in Nonlinear Analysis
Faria, Luiz F.O.; Miyagaki, Olimpio H.; Pereira, Fábio R.
We deal with the existence of solutions to a critical elliptic system involving the
fractional Laplacian operator. We consider the primitive of the nonlinearity interacting
with the spectrum of the operator. The one side resonant case is also considered.
Variational methods are used to obtain the existence, and our result improves earlier
results of the authors.
Faria, Luiz F.O.; Miyagaki, Olimpio H.; Pereira, Fábio R.
We deal with the existence of solutions to a critical elliptic system involving the
fractional Laplacian operator. We consider the primitive of the nonlinearity interacting
with the spectrum of the operator. The one side resonant case is also considered.
Variational methods are used to obtain the existence, and our result improves earlier
results of the authors.
Teng, Kaimin; Agarwal, Ravi P.
We consider the following quasilinear elliptic equation with convex and concave nonlinearities:
\begin{equation*}
-\Delta_p u-(\Delta_pu^2)u+V(x)|u|^{p-2}u=\lambda K(x) |u|^{q-2}u+\mu g(x,u),\quad \text{in }\mathbb{R}^N,
\end{equation*}
where $2\leq p< N$, $1< q< p$, $\lambda,\mu\in\mathbb{R}$, $V$ and $K$ are potential functions, and
$g\in C(\mathbb{R}^N\times\mathbb{R},\mathbb{R})$ is a continuous function.
Under some suitable conditions on $V,K$ and $g$, the existence of infinitely
many solutions is established.
Teng, Kaimin; Agarwal, Ravi P.
We consider the following quasilinear elliptic equation with convex and concave nonlinearities:
\begin{equation*}
-\Delta_p u-(\Delta_pu^2)u+V(x)|u|^{p-2}u=\lambda K(x) |u|^{q-2}u+\mu g(x,u),\quad \text{in }\mathbb{R}^N,
\end{equation*}
where $2\leq p< N$, $1< q< p$, $\lambda,\mu\in\mathbb{R}$, $V$ and $K$ are potential functions, and
$g\in C(\mathbb{R}^N\times\mathbb{R},\mathbb{R})$ is a continuous function.
Under some suitable conditions on $V,K$ and $g$, the existence of infinitely
many solutions is established.
Manuilov, Vladimir
Topological complexity for spaces was introduced by M. Farber as a minimal number
of continuity domains for motion planning algorithms. It turns out that this notion can be extended to
the case of not necessarily commutative $C^*$-algebras. Topological complexity for spaces is closely
related to the Lusternik-Schnirelmann category, for which we do not know any noncommutative extension,
so there is no hope to generalize the known estimation methods, but we are able to evaluate the topological
complexity for some very simple examples of noncommutative $C^*$-algebras.
Manuilov, Vladimir
Topological complexity for spaces was introduced by M. Farber as a minimal number
of continuity domains for motion planning algorithms. It turns out that this notion can be extended to
the case of not necessarily commutative $C^*$-algebras. Topological complexity for spaces is closely
related to the Lusternik-Schnirelmann category, for which we do not know any noncommutative extension,
so there is no hope to generalize the known estimation methods, but we are able to evaluate the topological
complexity for some very simple examples of noncommutative $C^*$-algebras.
Papageorgiou, Nikolaos S.; Vetro, Calogero; Vetro, Francesca
We consider a semilinear Robin problem driven by Laplacian plus an indefinite
and unbounded potential. The reaction function contains a concave term and a perturbation of arbitrary
growth. Using a variant of the symmetric mountain pass theorem, we show the existence of smooth nodal
solutions which converge to zero in $C^1(\overline{\Omega})$. If the coefficient of the concave term
is sign changing, then again we produce a sequence of smooth solutions converging to zero in
$C^1(\overline{\Omega})$, but we cannot claim that they are nodal.
Papageorgiou, Nikolaos S.; Vetro, Calogero; Vetro, Francesca
We consider a semilinear Robin problem driven by Laplacian plus an indefinite
and unbounded potential. The reaction function contains a concave term and a perturbation of arbitrary
growth. Using a variant of the symmetric mountain pass theorem, we show the existence of smooth nodal
solutions which converge to zero in $C^1(\overline{\Omega})$. If the coefficient of the concave term
is sign changing, then again we produce a sequence of smooth solutions converging to zero in
$C^1(\overline{\Omega})$, but we cannot claim that they are nodal.
Balanov, Zalman; Hooton, Edward; Murza, Adrian
Van der Pol equation (in short, vdP) as well as many its non-symmetric
generalizations (the so-called van der Pol-like oscillators (in short, vdPl)) serve as nodes in coupled
networks modeling real-life phenomena. Symmetric properties of periodic regimes of networks of vdP/vdPl
depend on symmetries of coupling. In this paper, we consider $N^3$ identical
vdP/vdPl oscillators arranged in a cubical lattice, where opposite faces are identified in the
same way as for a $3$-torus. Depending on which nodes impact the dynamics of a given node, we distinguish
between $\mathbb D_N \times \mathbb D_N \times \mathbb D_N$-equivariant systems and their
$\mathbb Z_N \times \mathbb Z_N \times \mathbb Z_N$-equivariant counterparts....
Balanov, Zalman; Hooton, Edward; Murza, Adrian
Van der Pol equation (in short, vdP) as well as many its non-symmetric
generalizations (the so-called van der Pol-like oscillators (in short, vdPl)) serve as nodes in coupled
networks modeling real-life phenomena. Symmetric properties of periodic regimes of networks of vdP/vdPl
depend on symmetries of coupling. In this paper, we consider $N^3$ identical
vdP/vdPl oscillators arranged in a cubical lattice, where opposite faces are identified in the
same way as for a $3$-torus. Depending on which nodes impact the dynamics of a given node, we distinguish
between $\mathbb D_N \times \mathbb D_N \times \mathbb D_N$-equivariant systems and their
$\mathbb Z_N \times \mathbb Z_N \times \mathbb Z_N$-equivariant counterparts....
Hu, Tingxi; Lu, Lu
We are concerned with the multiplicity of positive solutions for the
following Kirchhoff type problem:
\begin{equation*}
\begin{cases}
\displaystyle
- \bigg(\varepsilon ^2a + \varepsilon b\int_{{\mathbb{R}^3}}{|\nabla u{|^2}} dx \bigg)\Delta u
+ u = Q(x)|u|^{p-2}u,& x\in \mathbb{R}^3, \hfill \\
u \in H^1(\mathbb{R}^3), \quad u > 0, & x\in \mathbb{R}^3 ,
\end{cases}
\end{equation*}
where $\varepsilon> 0 $ is a parameter, $a, b> 0$ are constants, $p\in (2, 6)$, and
$Q\in C(\mathbb{R}^3)$ is a nonnegative function. We show how the profile of $Q$ affects the number
of positive solutions when $\varepsilon $ is sufficiently small.
Hu, Tingxi; Lu, Lu
We are concerned with the multiplicity of positive solutions for the
following Kirchhoff type problem:
\begin{equation*}
\begin{cases}
\displaystyle
- \bigg(\varepsilon ^2a + \varepsilon b\int_{{\mathbb{R}^3}}{|\nabla u{|^2}} dx \bigg)\Delta u
+ u = Q(x)|u|^{p-2}u,& x\in \mathbb{R}^3, \hfill \\
u \in H^1(\mathbb{R}^3), \quad u > 0, & x\in \mathbb{R}^3 ,
\end{cases}
\end{equation*}
where $\varepsilon> 0 $ is a parameter, $a, b> 0$ are constants, $p\in (2, 6)$, and
$Q\in C(\mathbb{R}^3)$ is a nonnegative function. We show how the profile of $Q$ affects the number
of positive solutions when $\varepsilon $ is sufficiently small.
Ding, Boyang; Ding, Changming
We aim to introduce the
generalized recurrence into the theory of impulsive semidynamical
systems. Similarly to Auslander's construction in [J. Auslander,
Generalized recurrence in dynamical systems,
Contrib. Differential Equations 3 (1964), 65-74], we
present two different characterizations, respectively, by Lyapunov
functions and higher prolongations. In fact, we show that if the
phase space is a locally compact separable metric space, then the
generalized recurrent set is the same as the quasi prolongational
recurrent set. Also, we see that
many new phenomena appear for the impulse effects in the
semidynamical system.
Ding, Boyang; Ding, Changming
We aim to introduce the
generalized recurrence into the theory of impulsive semidynamical
systems. Similarly to Auslander's construction in [J. Auslander,
Generalized recurrence in dynamical systems,
Contrib. Differential Equations 3 (1964), 65-74], we
present two different characterizations, respectively, by Lyapunov
functions and higher prolongations. In fact, we show that if the
phase space is a locally compact separable metric space, then the
generalized recurrent set is the same as the quasi prolongational
recurrent set. Also, we see that
many new phenomena appear for the impulse effects in the
semidynamical system.
Qi, Liangping; Yuan, Rong
We consider the existence of almost periodic solutions to differential equations
by using coincidence degree theory.
A new equivalent spectral condition for the compactness of integral operators on almost periodic function
spaces is established. It is shown that semigroup conditions are crucial in applications.
Qi, Liangping; Yuan, Rong
We consider the existence of almost periodic solutions to differential equations
by using coincidence degree theory.
A new equivalent spectral condition for the compactness of integral operators on almost periodic function
spaces is established. It is shown that semigroup conditions are crucial in applications.
Zhao, Xiaopeng
We study the global solvability and dynamical behaviour of the modified
Cahn-Hilliard equation with biological applications in the Sobolev space $H^1(\mathbb{R}^N)$.
Zhao, Xiaopeng
We study the global solvability and dynamical behaviour of the modified
Cahn-Hilliard equation with biological applications in the Sobolev space $H^1(\mathbb{R}^N)$.
Saavedra, Lorena; Tersian, Stepan
The aim of this paper is the study of existence of solutions for
nonlinear $2n^{\rm th}$-order difference equations involving
$p$-Laplacian. In the first part, the existence of a nontrivial
homoclinic solution for a discrete $p$-Laplacian problem is proved.
The proof is based on the mountain-pass theorem of Brezis and
Nirenberg. Then, we study the existence of multiple solutions for a
discrete $p$-Laplacian boundary value problem. In this case the
proof is based on the three critical points theorem of Averna and
Bonanno.
Saavedra, Lorena; Tersian, Stepan
The aim of this paper is the study of existence of solutions for
nonlinear $2n^{\rm th}$-order difference equations involving
$p$-Laplacian. In the first part, the existence of a nontrivial
homoclinic solution for a discrete $p$-Laplacian problem is proved.
The proof is based on the mountain-pass theorem of Brezis and
Nirenberg. Then, we study the existence of multiple solutions for a
discrete $p$-Laplacian boundary value problem. In this case the
proof is based on the three critical points theorem of Averna and
Bonanno.