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

On affinity of Peano type functions

Tomasz Słonka (2012)

Colloquium Mathematicae

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We show that if n is a positive integer and 2 , then for every positive integer m and for every real constant c > 0 there are functions f , . . . , f n + m : such that ( f , . . . , f n + m ) ( ) = 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 , . . . , f i ) ( y ) = y + w for y x + ( - c , c ) × n - 1 .

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

Alejandro Illanes (2016)

Commentationes Mathematicae Universitatis Carolinae

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A metric continuum X is said to be continuously homogeneous provided that for every two points p , q X there exists a continuous surjective function f : X 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

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

Perturbed nonlinear degenerate problems in N

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

Applicationes Mathematicae

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Via critical point theory we establish the existence and regularity of solutions for the quasilinear elliptic problem ⎧ d i v ( x , u ) + a ( x ) | u | p - 2 u = g ( x ) | u | p - 2 u + h ( x ) | u | s - 1 u in N ⎨ ⎩ u > 0, l i m | x | 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

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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.

Recent results on stationary critical Kirchhoff systems in closed manifolds

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

Séminaire Laurent Schwartz — EDP et applications

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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 3 . The critical Kirchhoff systems we consider are written as a + b j = 1 p M | u j | 2 d v g Δ g u i + j = 1 p A i j u j = U 2 - 2 u i for all i = 1 , , p , where Δ g is the Laplace-Beltrami operator, A is a C 1 -map from M into the space M s p ( ) of symmetric p × p matrices with real entries, the A i j ’s are the components of A , U = ( u 1 , , u p ) , | U | : M is the Euclidean norm of U , 2 = 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

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On the unit disk B 1 2 we study the Moser-Trudinger functional E ( u ) = B 1 e u 2 - 1 d x , u H 0 1 ( B 1 ) and its restrictions E | M Λ , where M Λ : = { u 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 , then Λ k 4 π , and u k 0 weakly in H 0 1 ( B 1 ) and strongly in C loc 1 ( 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.

Nonnormality of remainders of some topological groups

Aleksander V. Arhangel&amp;#039;skii, J. van Mill (2016)

Commentationes Mathematicae Universitatis Carolinae

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It is known that every remainder of a topological group is Lindelöf or pseudocompact. Motivated by this result, we study in this paper when a topological group G has a normal remainder. In a previous paper we showed that under mild conditions on G , the Continuum Hypothesis implies that if the Čech-Stone remainder G * of G is normal, then it is Lindelöf. Here we continue this line of investigation, mainly for the case of precompact groups. We show that no pseudocompact group, whose weight...

Scaling limit and cube-root fluctuations in SOS surfaces above a wall

Pietro Caputo, Eyal Lubetzky, Fabio Martinelli, Allan Sly, Fabio Lucio Toninelli (2016)

Journal of the European Mathematical Society

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Consider the classical ( 2 + 1 ) -dimensional Solid-On-Solid model above a hard wall on an L × L box of 2 . The model describes a crystal surface by assigning a non-negative integer height η x to each site x in the box and 0 heights to its boundary. The probability of a surface configuration η is proportional to exp ( - β ( η ) ) , where β is the inverse-temperature and ( η ) sums the absolute values of height differences between neighboring sites. We give a full description of the shape of the SOS surface for low enough...

Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity

Olivier Rey, Juncheng Wei (2005)

Journal of the European Mathematical Society

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We show that the critical nonlinear elliptic Neumann problem Δ u μ u + u 7 / 3 = 0 in Ω , u > 0 in Ω , u ν = 0 on Ω , where Ω is a bounded and smooth domain in 5 , has arbitrarily many solutions, provided that μ > 0 is small enough. More precisely, for any positive integer K , there exists μ K > 0 such that for 0 < μ < μ K , the above problem has a nontrivial solution which blows up at K interior points in Ω , as μ 0 . The location of the blow-up points is related to the domain geometry. The solutions are obtained as critical points of some finite-dimensional...

On the classification of inverse limits of tent maps

Louis Block, Slagjana Jakimovik, Lois Kailhofer, James Keesling (2005)

Fundamenta Mathematicae

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Let f s and f t be tent maps on the unit interval. In this paper we give a new proof of the fact that if the critical points of f s and f t are periodic and the inverse limit spaces ( I , f s ) and ( I , f t ) are homeomorphic, then s = t. This theorem was first proved by Kailhofer. The new proof in this paper simplifies the proof of Kailhofer. Using the techniques of the paper we are also able to identify certain isotopies between homeomorphisms on the inverse limit space.

A compactness result for polyharmonic maps in the critical dimension

Shenzhou Zheng (2016)

Czechoslovak Mathematical Journal

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For n = 2 m 4 , let Ω n be a bounded smooth domain and 𝒩 L a compact smooth Riemannian manifold without boundary. Suppose that { u k } W m , 2 ( Ω , 𝒩 ) is a sequence of weak solutions in the critical dimension to the perturbed m -polyharmonic maps d d t | t = 0 E m ( Π ( u + t ξ ) ) = 0 with Φ k 0 in ( W m , 2 ( Ω , 𝒩 ) ) * and u k u weakly in W m , 2 ( Ω , 𝒩 ) . Then u is an m -polyharmonic map. In particular, the space of m -polyharmonic maps is sequentially compact for the weak- W m , 2 topology.

Singular arc-like continua

Tadeusz Maćkowiak

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CONTENTSIntroduction.......................................................................................................51. Preliminaries.................................................................................................6  A. Mappings....................................................................................................6  B. Arc-like continua.........................................................................................8  C. Pseudosuspensions...................................................................................8  D....