Study of the critical points at infinity arising from the failure of the Palais-Smale condition for 𝑛-body type problems

1999 ◽  
Vol 138 (658) ◽  
pp. 0-0 ◽  
Author(s):  
Hasna Riahi
Keyword(s):  
Critical Points ◽  
Body Type ◽  
Critical Points At Infinity ◽  
Palais Smale Condition

scholarly journals Multiplicity of Solutions for Nonlocal Elliptic System of(p,q)-Kirchhoff Type

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Bitao Cheng ◽  
Xian Wu ◽  
Jun Liu
Keyword(s):  
Elliptic System ◽  
Critical Points ◽  
Three Solutions ◽  
Multiplicity Of Solutions ◽  
Periodic Condition ◽  
Kirchhoff Type ◽  
Three Critical Points Theorem ◽  
Palais Smale Condition

This paper is concerned with the following nonlocal elliptic system of (p,q)-Kirchhoff type−[M1(∫Ω|∇u|p)]p−1Δpu=λFu(x,u,v), in Ω,−[M2(∫Ω|∇v|q)]q−1Δqv=λFv(x,u,v), in Ω,u=v=0, on∂Ω.Under bounded condition onMand some novel and periodic condition onF, some new results of the existence of two solutions and three solutions of the above mentioned nonlocal elliptic system are obtained by means of Bonanno's multiple critical points theorems without the Palais-Smale condition and Ricceri's three critical points theorem, respectively.

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.

Multiple solutions for elliptic resonant problems

2008 ◽  
Vol 138 (6) ◽  
pp. 1281-1289 ◽  
Author(s):  
Shibo Liu
Keyword(s):  
Critical Points ◽  
Multiple Solutions ◽  
Careful Analysis ◽  
Variational Functional ◽  
Three Critical Points Theorem ◽  
Semilinear Elliptic ◽  
Palais Smale Condition

Two non-trivial solutions for semilinear elliptic resonant problems are obtained via the Lyapunov—Schmidt reduction and the three-critical-points theorem. The difficulty that the variational functional does not satisfy the Palais—Smale condition is overcome by taking advantage of the reduction and a careful analysis of the reduced functional.

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.

scholarly journals Solutions for a singular elliptic problem involving the p(x)-Laplacian

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 4841-4850 ◽  
Author(s):  
Khanghahi Mahdavi ◽  
A. Razani
Keyword(s):  
Bounded Domain ◽  
Critical Points ◽  
Elliptic Problem ◽  
Energy Functional ◽  
Laplacian Operator ◽  
Palais Smale Condition

Here, a singular elliptic problem involving p(x)-Laplacian operator in a bounded domain in RN is considered. Due to this, the existence of critical points for the energy functional which is unbounded below and satisfies the Palais-Smale condition are proved.

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