scholarly journals Critical points at infinity in the variational calculus

1988 ◽  
pp. 1-29 ◽  
Author(s):  
A. Bahri
Keyword(s):  
Critical Points ◽  
Variational Calculus ◽  
Critical Points At Infinity

scholarly journals Global Phase Portrait and Bifurcation Diagram for a Multi-Parametric Linear System

2020 ◽  
Vol 55 (4) ◽  
Author(s):  
Jorge Rodríguez Contreras ◽  
Alberto Reyes Linero ◽  
Juliana Vargas Sánchez
Keyword(s):  
Linear System ◽  
Critical Point ◽  
Phase Portrait ◽  
Bifurcation Diagram ◽  
Critical Points ◽  
Global Dynamics ◽  
Parametric Surfaces ◽  
Critical Points At Infinity ◽  
Global Phase Portrait ◽  
The Stability

The goal of this article is to conduct a global dynamics study of a linear multiparameter system (real parameters (a,b,c) in R^3); for this, we take the different changes that these parameters present. First, we find the different parametric surfaces in which the space is divided, where the stability of the critical point is defined; we then create a bifurcation diagram to classify the different bifurcations that appear in the system. Finally, we determine and classify the critical points at infinity, considering the canonical shape of the Poincaré sphere, and thus, obtain a global phase portrait of the multiparametric linear system.

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.

PERTURBATION THEOREMS FOR FRACTIONAL CRITICAL EQUATIONS ON BOUNDED DOMAINS

2020 ◽  
pp. 1-20 ◽  
Author(s):  
AZEB ALGHANEMI ◽  
HICHEM CHTIOUI
Keyword(s):  
Bounded Domain ◽  
Laplace Operator ◽  
Critical Points ◽  
Existence Theorems ◽  
Variational Problems ◽  
Bounded Domains ◽  
Critical Problem ◽  
Critical Points At Infinity ◽  
Fractional Laplace Operator

We consider the fractional critical problem $A_{s}u=K(x)u^{(n+2s)/(n-2s)},u>0$ in $\unicode[STIX]{x1D6FA},u=0$ on $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$ , where $A_{s},s\in (0,1)$ , is the fractional Laplace operator and $K$ is a given function on a bounded domain $\unicode[STIX]{x1D6FA}$ of $\mathbb{R}^{n},n\geq 2$ . This is based on A. Bahri’s theory of critical points at infinity in Bahri [Critical Points at Infinity in Some Variational Problems, Pitman Research Notes in Mathematics Series, 182 (Longman Scientific & Technical, Harlow, 1989)]. We prove Bahri’s estimates in the fractional setting and we provide existence theorems for the problem when $K$ is close to 1.

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