On two problems studied by A. Ambrosetti

David Arcoya; José Carmona

Displaying similar documents to “On two problems studied by A. Ambrosetti”

On the hyperspace $C_{n} (X) / {C_{n}}_{K} (X)$

José G. Anaya, Enrique Castañeda-Alvarado, José A. Martínez-Cortez (2021)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Let $X$ be a continuum and $n$ a positive integer. Let $C_{n} (X)$ be the hyperspace of all nonempty closed subsets of $X$ with at most $n$ components, endowed with the Hausdorff metric. For $K$ compact subset of $X$ , define the hyperspace ${C_{n}}_{K} (X) = {A \in C_{n} (X) : K \subset A}$ . In this paper, we consider the hyperspace $C_{K}^{n} (X) = C_{n} (X) / {C_{n}}_{K} (X)$ , which can be a tool to study the space $C_{n} (X)$ . We study this hyperspace in the class of finite graphs and in general, we prove some properties such as: aposyndesis, local connectedness, arcwise disconnectedness, and contractibility. ...

On affinity of Peano type functions

Tomasz Słonka (2012)

Colloquium Mathematicae

Similarity:

We show that if n is a positive integer and $2^{ℵ ₀} \leq ℵ ₙ$ , then for every positive integer m and for every real constant c > 0 there are functions $f ₁, . . ., f_{n + m} : ℝ ⁿ \to ℝ$ such that $(f ₁, . . ., f_{n + m}) (ℝ ⁿ) = ℝ^{n + m}$ and for every x ∈ ℝⁿ there exists a strictly increasing sequence (i₁,...,iₙ) of numbers from 1,...,n+m and a w ∈ ℤⁿ such that $(f_{i ₁}, . . ., f_{i ₙ}) (y) = y + w$ for $y \in x + (- c, c) \times ℝ^{n - 1}$ .

$R_{0}$ and $R_{1}$ topological spaces

C. Dorsett (1978)

Matematički Vesnik

Similarity:

On $R_{0}$ and $R_{1}$ topological spaces

D. Janković (1981)

Matematički Vesnik

Similarity:

A note on $R_{0}$ and $R_{1}$ topological spaces

M. R. Žižović (1986)

Matematički Vesnik

Similarity:

Making holes in the cone, suspension and hyperspaces of some continua

José G. Anaya, Enrique Castañeda-Alvarado, Alejandro Fuentes-Montes de Oca, Fernando Orozco-Zitli (2018)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

A connected topological space $Z$ is unicoherent provided that if $Z = A \cup B$ where $A$ and $B$ are closed connected subsets of $Z$ , then $A \cap B$ is connected. Let $Z$ be a unicoherent space, we say that $z \in Z$ makes a hole in $Z$ if $Z - {z}$ is not unicoherent. In this work the elements that make a hole to the cone and the suspension of a metric space are characterized. We apply this to give the classification of the elements of hyperspaces of some continua that make them hole.

A continuum $X$ such that $C (X)$ is not continuously homogeneous

Alejandro Illanes (2016)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

A metric continuum $X$ is said to be continuously homogeneous provided that for every two points $p, q \in X$ there exists a continuous surjective function $f : X \to X$ such that $f (p) = q$ . Answering a question by W.J. Charatonik and Z. Garncarek, in this paper we show a continuum $X$ such that the hyperspace of subcontinua of $X$ , $C (X)$ , is not continuously homogeneous.

Location of the critical points of certain polynomials

Somjate Chaiya, Aimo Hinkkanen (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

Let $𝔻$ denote the unit disk ${z : | z | < 1}$ in the complex plane $ℂ$ . In this paper, we study a family of polynomials $P$ with only one zero lying outside $\bar{𝔻}$ . We establish criteria for $P$ to satisfy implying that each of $P$ and $P^{'}$ has exactly one critical point outside $\bar{𝔻}$ .

Perturbed nonlinear degenerate problems in $ℝ^{N}$

A. El Khalil, S. El Manouni, M. Ouanan (2009)

Applicationes Mathematicae

Similarity:

Via critical point theory we establish the existence and regularity of solutions for the quasilinear elliptic problem ⎧ $d i v (x, \nabla u) + {a (x) | u |}^{p - 2} u = {g (x) | u |}^{p - 2} u + h (x) {| u |}^{s - 1} u$ in $ℝ^{N}$ ⎨ ⎩ u > 0, $l i m_{| x | \to \infty} u (x) = 0$ , where 1 < p < N; a(x) is assumed to satisfy a coercivity condition; h(x) and g(x) are not necessarily bounded but satisfy some integrability restrictions.

Extending generalized Whitney maps

Ivan Lončar (2017)

Archivum Mathematicum

Similarity:

For metrizable continua, there exists the well-known notion of a Whitney map. If $X$ is a nonempty, compact, and metric space, then any Whitney map for any closed subset of $2^{X}$ can be extended to a Whitney map for $2^{X}$ [3, 16.10 Theorem]. The main purpose of this paper is to prove some generalizations of this theorem.

Continuous images of Lindelöf $p$ -groups, $σ$ -compact groups, and related results

Aleksander V. Arhangel'skii (2019)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

It is shown that there exists a $σ$ -compact topological group which cannot be represented as a continuous image of a Lindelöf $p$ -group, see Example 2.8. This result is based on an inequality for the cardinality of continuous images of Lindelöf $p$ -groups (Theorem 2.1). A closely related result is Corollary 4.4: if a space $Y$ is a continuous image of a Lindelöf $p$ -group, then there exists a covering $γ$ of $Y$ by dyadic compacta such that $| γ | \leq 2^{ω}$ . We also show that if a homogeneous compact space $Y$ is...

Recent results on stationary critical Kirchhoff systems in closed manifolds

Emmanuel Hebey, Pierre-Damien Thizy (2013-2014)

Séminaire Laurent Schwartz — EDP et applications

Similarity:

We report on results we recently obtained in Hebey and Thizy [11, 12] for critical stationary Kirchhoff systems in closed manifolds. Let $(M^{n}, g)$ be a closed $n$ -manifold, $n \geq 3$ . The critical Kirchhoff systems we consider are written as $(a + b \sum_{j = 1}^{p} \int_{M} {| \nabla u_{j} |}^{2} d v_{g}) Δ_{g} u_{i} + \sum_{j = 1}^{p} A_{i j} u_{j} = {|U|}^{2^{☆} - 2} u_{i}$ for all $i = 1, \dots, p$ , where $Δ_{g}$ is the Laplace-Beltrami operator, $A$ is a $C^{1}$ -map from $M$ into the space $M_{s}^{p} (ℝ)$ of symmetric $p \times p$ matrices with real entries, the $A_{i j}$ ’s are the components of $A$ , $U = (u_{1}, \dots, u_{p})$ , $| U | : M \to ℝ$ is the Euclidean norm of $U$ , $2^{☆} = \frac{2 n}{n - 2}$ is the critical Sobolev exponent, and...

Critical points of the Moser-Trudinger functional on a disk

Andrea Malchiodi, Luca Martinazzi (2014)

Journal of the European Mathematical Society

Similarity:

On the unit disk $B_{1} \subset ℝ^{2}$ we study the Moser-Trudinger functional $E (u) = \int_{B_{1}} (e^{u^{2}} - 1) d x, u \in H_{0}^{1} (B_{1})$ and its restrictions ${E |}_{M_{Λ}}$ , where $M_{Λ} : = {u \in H_{0}^{1} (B_{1}) {: ∥ u ∥}_{H_{0}^{1}}^{2} = Λ}$ for $Λ > 0$ . We prove that if a sequence $u_{k}$ of positive critical points of ${E |}_{M_{Λ_{k}}}$ (for some $Λ_{k} > 0$ ) blows up as $k \to \infty$ , then $Λ_{k} \to 4 π$ , and $u_{k} \to 0$ weakly in $H_{0}^{1} (B_{1})$ and strongly in $C_{loc}^{1} ({\bar{B}}_{1} ∖ {0})$ . Using this fact we also prove that when $Λ$ is large enough, then ${E |}_{M_{Λ}}$ has no positive critical point, complementing previous existence results by Carleson-Chang, M. Struwe and Lamm-Robert-Struwe.