scholarly journals Prescribing the Scalar Curvature under Minimal Boundary Conditions on the Half Sphere

2002 ◽  
Vol 2 (2) ◽  
Author(s):  
Mohamed Ben Ayed ◽  
Khalil El Mehdi ◽  
Mohameden Ould Ahmedou
Keyword(s):  
Boundary Conditions ◽  
Variational Problem ◽  
Scalar Curvature ◽  
Critical Points ◽  
Existence Results ◽  
Topological Methods ◽  
Critical Points At Infinity

AbstractThis paper is devoted to the problem of prescribing the scalar curvature under zero boundary conditions. Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, we prove some existence results on the standard half sphere.

Theory of “Critical Points at Infinity” and a Resonant Singular Liouville-Type Equation

2017 ◽  
Vol 17 (1) ◽  
Author(s):  
Mohameden Ahmedou ◽  
Mohamed Ben Ayed
Keyword(s):  
Variational Problem ◽  
Critical Point ◽  
Critical Points ◽  
Critical Point Theory ◽  
Type Equation ◽  
Point Theory ◽  
Existence Results ◽  
Liouville Type ◽  
Resonant Case ◽  
Critical Points At Infinity

AbstractWe consider the following Liouville-type equation on domains ofwhereUsing some dynamical and topological tools from the “critical point theory at infinity” of Bahri, we study the critical points at infinity of the related variational problem. Then we derive from our analysis some existence results in the so-called resonant case, that is, when the parameter ϱ is of the form

scholarly journals Variational and Topological Methods for a Class of Nonlinear Equations which Involves a Duality Mapping

2020 ◽  
pp. 56-71
Author(s):  
Jenica Cringanu
Keyword(s):  
Banach Space ◽  
Variational Method ◽  
Critical Points ◽  
Mountain Pass Theorem ◽  
Growth Conditions ◽  
Existence Results ◽  
Duality Mapping ◽  
Topological Methods ◽  
Carathéodory Function ◽  
Direct Variational Method

The purpose of this paper is to show the existence results for the following abstract equation Jpu = Nfu,where Jp is the duality application on a real reflexive and smooth X Banach space, that corresponds to the gauge function φ(t) = tp-1, 1 < p < ∞. We assume that X is compactly imbedded in Lq(Ω), where Ω is a bounded domain in RN, N ≥ 2, 1 < q < p∗, p∗ is the Sobolev conjugate exponent.Nf : Lq(Ω) → Lq′(Ω), 1/q + 1/q′ = 1, is the Nemytskii operator that Caratheodory function generated by a f : Ω × R → R which satisfies some growth conditions. We use topological methods (via Leray-Schauder degree), critical points methods (the Mountain Pass theorem) and a direct variational method to prove the existence of the solutions for the equation Jpu = Nfu.

Prescribing the Scalar Curvature Problem on Three and Four Manifolds

2003 ◽  
Vol 3 (4) ◽  
Author(s):  
Hichem Chtioui
Keyword(s):  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Critical Points ◽  
New Class ◽  
Critical Points At Infinity ◽  
Four Manifolds ◽  
Curvature Problem

AbstractThis paper is devoted to the prescribed scalar curvature problem on 3 and 4- dimensional Riemannian manifolds. We give a new class of functionals which can be realized as scalar curvature. Our proof uses topological arguments and the tools of the theory of the critical points at infinity.

β-flatness condition in CR spheres multiplicity results

2020 ◽  
Vol 31 (03) ◽  
pp. 2050023
Author(s):  
Najoua Gamara ◽  
Boutheina Hafassa ◽  
Akrem Makni
Keyword(s):  
Lower Bound ◽  
Scalar Curvature ◽  
Critical Points ◽  
Number Of Solutions ◽  
Multiplicity Results ◽  
Type Formula ◽  
Critical Points At Infinity ◽  
Cauchy Riemann

We give multiplicity results for the problem of prescribing the scalar curvature on Cauchy–Riemann spheres under [Formula: see text]-flatness condition. To find a lower bound for the number of solutions, we use Bahri’s methods based on the theory of critical points at infinity and a Poincaré–Hopf-type formula.

scholarly journals Prescribing scalar curvatures: non compactness versus critical points at infinity

2019 ◽  
Vol 4 (1) ◽  
pp. 51-82 ◽  
Author(s):  
Martin Mayer
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Critical Points ◽  
Gradient Flow ◽  
Slight Modification ◽  
Positive Function ◽  
Flow Lines ◽  
Critical Points At Infinity

Abstract We illustrate an example of a generic, positive function K on a Riemannian manifold to be conformally prescribed as the scalar curvature, for which the corresponding Yamabe type L2-gradient flow exhibits non compact flow lines, while a slight modification of it is compact.

scholarly journals Conformal scalar curvature equation on Sn: Functions with two close critical points (Twin Pseudo-Peaks)

2018 ◽  
Vol 20 (05) ◽  
pp. 1750051
Author(s):  
Man Chun Leung ◽  
Feng Zhou
Keyword(s):  
Scalar Curvature ◽  
Critical Points ◽  
Reduction Method ◽  
Existence Results ◽  
The Other ◽  
Curvature Equation ◽  
Two Bubbles

By using the Lyapunov–Schmidt reduction method without perturbation, we consider existence results for the conformal scalar curvature on [Formula: see text] ([Formula: see text]) when the prescribed function (after being projected to [Formula: see text]) has two close critical points, which have the same value (positive), equal “flatness” (“twin”; flatness [Formula: see text]), and exhibit maximal behavior in certain directions (“pseudo-peaks”). The proof relies on a balance between the two main contributions to the reduced functional — one from the critical points and the other from the interaction of the two bubbles.

scholarly journals Prescribing Morse Scalar Curvatures: Blow-Up Analysis

2020 ◽  
Author(s):  
Andrea Malchiodi ◽  
Martin Mayer
Keyword(s):  
Critical Points ◽  
Blow Up ◽  
Finite Energy ◽  
Existence Results ◽  
Critical Regime ◽  
Geometric Problem ◽  
Critical Points At Infinity ◽  
Blow Up Analysis ◽  
Blowing Up ◽  
Subcritical Solutions

Abstract We study finite-energy blow-ups for prescribed Morse scalar curvatures in both the subcritical and the critical regime. After general considerations on Palais–Smale sequences, we determine precise blow-up rates for subcritical solutions: in particular the possibility of tower bubbles is excluded in all dimensions. In subsequent papers, we aim to establish the sharpness of this result, proving a converse existence statement, together with a one-to-one correspondence of blowing-up subcritical solutions and critical points at infinity. This analysis will be then applied to deduce new existence results for the geometric problem.

Topological Methods for Boundary Mean Curvature Problem on Bn

2014 ◽  
Vol 14 (2) ◽  
Author(s):  
Mohammed Ali Al-Ghamdi ◽  
Hichem Chtioui ◽  
Khadijah Sharaf
Keyword(s):  
Critical Points ◽  
Mean Curvature ◽  
Topological Method ◽  
Topological Methods ◽  
Critical Points At Infinity ◽  
The Mean ◽  
Curvature Problem ◽  
Boundary Mean Curvature

AbstractUsing an algebraic topological method and the tools of the theory of the critical points at infinity, we provide a variety of classes of functions that can be realized as the mean curvature on the boundary of the the n-dimensional balls.

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