Real Koebe principle
Weixiao Shen, Michael Todd (2005)
Fundamenta Mathematicae
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We prove a version of the real Koebe principle for interval (or circle) maps with non-flat critical points.
Weixiao Shen, Michael Todd (2005)
Fundamenta Mathematicae
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We prove a version of the real Koebe principle for interval (or circle) maps with non-flat critical points.
Lan Zeng, Chun Lei Tang (2016)
Annales Polonici Mathematici
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We consider the following Kirchhoff type problem involving a critical nonlinearity: ⎧ in Ω, ⎨ ⎩ u = 0 on ∂Ω, where (N ≥ 3) is a smooth bounded domain with smooth boundary ∂Ω, a > 0, b ≥ 0, and 0 < m < 2/(N-2). Under appropriate assumptions on f, we show the existence of a positive ground state solution via the variational method.
Mike Todd (2007)
Fundamenta Mathematicae
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We obtain estimates for derivative and cross-ratio distortion for (any η > 0) unimodal maps with non-flat critical points. We do not require any “Schwarzian-like” condition. For two intervals J ⊂ T, the cross-ratio is defined as the value B(T,J): = (|T| |J|)/(|L| |R|) where L,R are the left and right connected components of T∖J respectively. For an interval map g such that is a diffeomorphism, we consider the cross-ratio distortion to be B(g,T,J): = B(g(T),g(J))/B(T,J). We prove...
David E. Edmunds, Donato Fortunato, Enrico Jannelli (1989)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
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In this paper we consider a nonlinear elliptic equation with critical growth for the operator in a bounded domain . We state some existence results when . Moreover, we consider , expecially when is a ball in .
Edson de Faria, Welington de Melo (1999)
Journal of the European Mathematical Society
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We prove that two critical circle maps with the same rotation number in a special set are conjugate for some provided their successive renormalizations converge together at an exponential rate in the sense. The set has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of critical circle maps with the same rotation number that are not conjugate for any . The class of rotation numbers for which such examples exist...
Lois Kailhofer (2003)
Fundamenta Mathematicae
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We work within the one-parameter family of symmetric tent maps, where the slope is the parameter. Given two such tent maps , with periodic critical points, we show that the inverse limit spaces and are not homeomorphic when a ≠ b. To obtain our result, we define topological substructures of a composant, called “wrapping points” and “gaps”, and identify properties of these substructures preserved under a homeomorphism.
Vitaly Moroz, Cyrill B. Muratov (2014)
Journal of the European Mathematical Society
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We study the leading order behaviour of positive solutions of the equation , where , and when is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of , and . The behavior of solutions depends sensitively on whether is less, equal or bigger than the critical Sobolev exponent . For the solution asymptotically coincides with the solution of the equation in which the last term is absent. For the solution asymptotically...
David E. Edmunds, Donato Fortunato, Enrico Jannelli (1989)
Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti
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In this paper we consider a nonlinear elliptic equation with critical growth for the operator in a bounded domain . We state some existence results when . Moreover, we consider , expecially when is a ball in .
Michael Usher (2014)
Annales de la faculté des sciences de Toulouse Mathématiques
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For a Morse function on a compact oriented manifold , we show that has more critical points than the number required by the Morse inequalities if and only if there exists a certain class of link in whose components have nontrivial linking number, such that the minimal value of on one of the components is larger than its maximal value on the other. Indeed we characterize the precise number of critical points of in terms of the Betti numbers of and the behavior of with respect...
Manuel del Pino, Monica Musso, Frank Pacard (2010)
Journal of the European Mathematical Society
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The role of the second critical exponent , the Sobolev critical exponent in one dimension less, is investigated for the classical Lane–Emden–Fowler problem , under zero Dirichlet boundary conditions, in a domain in with bounded, smooth boundary. Given , a geodesic of the boundary with negative inner normal curvature we find that for , there exists a solution such that converges weakly to a Dirac measure on as , provided that is nondegenerate in the sense of second...
Benedetta Noris, Hugo Tavares, Susanna Terracini, Gianmaria Verzini (2012)
Journal of the European Mathematical Society
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In the recent literature, the phenomenon of phase separation for binary mixtures of Bose–Einstein condensates can be understood, from a mathematical point of view, as governed by the asymptotic limit of the stationary Gross–Pitaevskii system , as the interspecies scattering length goes to . For this system we consider the associated energy functionals , with -mass constraints, which limit (as ) is strongly irregular. For such functionals, we construct multiple critical points...
M.S. Shahrokhi-Dehkordi (2017)
Communications in Mathematics
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Let be a bounded starshaped domain and consider the -Laplacian problem where is a positive parameter, , and is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the -Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.
Werner Georg Nowak (2017)
Communications in Mathematics
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In the problem of (simultaneous) Diophantine approximation in (in the spirit of Hurwitz’s theorem), lower bounds for the critical determinant of the special three-dimensional body play an important role; see [1], [6]. This article deals with estimates from below for the critical determinant of more general star bodies where is any positive constant. These are obtained by inscribing into either a double cone, or an ellipsoid, or a double paraboloid, depending on the size of...
Zbigniew Jelonek (2004)
Banach Center Publications
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Let X ⊂ kⁿ be a smooth affine variety of dimension n-r and let be a polynomial dominant mapping. It is well-known that the mapping f is a locally trivial fibration outside a small closed set B(f). It can be proved (using a general Fibration Theorem of Rabier) that the set B(f) is contained in the set K(f) of generalized critical values of f. In this note we study the Rabier function. We give a few equivalent expressions for this function, in particular we compare this function with...
Jaeyoung Byeon, Kazunaga Tanaka (2013)
Journal of the European Mathematical Society
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We consider a singularly perturbed elliptic equation on , , where for any . The singularly perturbed problem has corresponding limiting problems on , . Berestycki-Lions found almost necessary and sufficient conditions on nonlinearity for existence of a solution of the limiting problem. There have been endeavors to construct solutions of the singularly perturbed problem concentrating around structurally stable critical points of potential under possibly general conditions...
Somjate Chaiya, Aimo Hinkkanen (2013)
Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica
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Let denote the unit disk in the complex plane . In this paper, we study a family of polynomials with only one zero lying outside . We establish criteria for to satisfy implying that each of and has exactly one critical point outside .
J. Chabrowski, Shusen Yan (2002)
Colloquium Mathematicae
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We consider the Neumann problem for the equation , u ∈ H¹(Ω), where Q is a positive and continuous coefficient on Ω̅ and λ is a parameter between two consecutive eigenvalues and . Applying a min-max principle based on topological linking we prove the existence of a solution.
Andrea Malchiodi, Luca Martinazzi (2014)
Journal of the European Mathematical Society
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On the unit disk we study the Moser-Trudinger functional and its restrictions , where for . We prove that if a sequence of positive critical points of (for some ) blows up as , then , and weakly in and strongly in . Using this fact we also prove that when is large enough, then has no positive critical point, complementing previous existence results by Carleson-Chang, M. Struwe and Lamm-Robert-Struwe.
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 ⎧ in ⎨ ⎩ u > 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.
Olivier Rey, Juncheng Wei (2005)
Journal of the European Mathematical Society
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We show that the critical nonlinear elliptic Neumann problem in , in , on , where is a bounded and smooth domain in , has arbitrarily many solutions, provided that is small enough. More precisely, for any positive integer , there exists such that for , the above problem has a nontrivial solution which blows up at interior points in , as . The location of the blow-up points is related to the domain geometry. The solutions are obtained as critical points of some finite-dimensional...
Y. Akdim, J. Bennouna, M. Mekkour, H. Redwane (2012)
Applicationes Mathematicae
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We study the problem ∂b(x,u)/∂t - div(a(x,t,u,Du)) + H(x,t,u,Du) = μ in Q = Ω×(0,T), in Ω, u = 0 in ∂Ω × (0,T). The main contribution of our work is to prove the existence of a renormalized solution without the sign condition or the coercivity condition on H(x,t,u,Du). The critical growth condition on H is only with respect to Du and not with respect to u. The datum μ is assumed to be in and b(x,u₀) ∈ L¹(Ω).
Viviane Baladi, Daniel Smania (2012)
Annales scientifiques de l'École Normale Supérieure
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We consider families of unimodal maps whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure of depends differentiably on , as a distribution of order . The proof uses transfer operators on towers whose level boundaries are mollified via smooth cutoff functions, in order to avoid artificial discontinuities. We give a new representation of for a Benedicks-Carleson map , in terms of a single smooth function and the...
Man-Ho Ho (2014)
Annales mathématiques Blaise Pascal
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In this note we prove some results in flat and differential -theory. The first one is a proof of the compatibility of the differential topological index and the flat topological index by a direct computation. The second one is the explicit isomorphisms between Bunke-Schick differential -theory and Freed-Lott differential -theory.
Djairo Guedes de Figueiredo, Jean-Pierre Gossez, Pedro Ubilla (2006)
Journal of the European Mathematical Society
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We study the existence, nonexistence and multiplicity of positive solutions for the family of problems , , where is a bounded domain in , and is a parameter. The results include the well-known nonlinearities of the Ambrosetti–Brezis–Cerami type in a more general form, namely , where . The coefficient is assumed to be nonnegative but is allowed to change sign, even in the critical case. The notions of local superlinearity and local sublinearity introduced in [9] are essential...
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 be a closed -manifold, . The critical Kirchhoff systems we consider are written as for all , where is the Laplace-Beltrami operator, is a -map from into the space of symmetric matrices with real entries, the ’s are the components of , , is the Euclidean norm of , is the critical Sobolev exponent, and...
Didier D&#039;Acunto, Vincent Grandjean (2005)
Annales Polonici Mathematici
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Let f: ℝⁿ → ℝ be a C² semialgebraic function and let c be an asymptotic critical value of f. We prove that there exists a smallest rational number such that |x|·|∇f| and are separated at infinity. If c is a regular value and , then f is a locally trivial fibration over c, and the trivialisation is realised by the flow of the gradient field of f.