Minimum action solutions of non-linear elliptic equations in unbounded domains containing a plane

1986 ◽  
Vol 102 (3-4) ◽  
pp. 327-343 ◽  
Author(s):  
Otared Kavian
Keyword(s):  
Elliptic Equation ◽  
Continuous Function ◽  
Bounded Domain ◽  
Elliptic Equations ◽  
Sufficient Conditions ◽  
Unbounded Domains ◽  
Necessary And Sufficient Conditions ◽  
Smooth Bounded Domain ◽  
Non Linear ◽  
Necessary And Sufficient

SynopsisLet d ≧ 1 be an integer and ω ⊂ℝd a smooth bounded domain and consider the elliptic equation − Δu = g(u) on Ω = ℝ2 × ω. We prove that under (almost) necessary and sufficient conditions on the continuous function g: ℝm→ ℝm the above equation has a minimum-action solution.

scholarly journals Completeness Theorems for Elliptic Equations of Higher Order with Constant Coefficients

2007 ◽  
Vol 14 (1) ◽  
pp. 81-97
Author(s):  
Alberto Cialdea
Keyword(s):  
Elliptic Equation ◽  
Bounded Domain ◽  
Elliptic Equations ◽  
Complete System ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Polynomial Solutions ◽  
Constant Coefficients ◽  
Necessary And Sufficient ◽  
Completeness Theorems

Abstract Let {ω𝑘 } be a complete system of polynomial solutions of the elliptic equation ∑|α|⩽2𝑚 aα 𝐷 α 𝑢 = 0, aα being real constants. We give necessary and sufficient conditions for the completeness of the system in [𝐿𝑝(∂Ω)]𝑚, where Ω ⊂ is a bounded domain such that is connected and ∂Ω ∈ 𝐶1.

scholarly journals Sharp Liouville Theorems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Salvador Villegas
Keyword(s):  
Elliptic Equations ◽  
Sufficient Conditions ◽  
Unbounded Domains ◽  
Necessary And Sufficient Conditions ◽  
Qualitative Properties ◽  
Liouville Theorems ◽  
Necessary And Sufficient

AbstractConsider the equation {\operatorname{div}(\varphi^{2}\nabla\sigma)=0} in {\mathbb{R}^{N}}, where {\varphi>0}. Berestycki, Caffarelli and Nirenberg proved in [H. Berestycki, L. Caffarelli and L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 25 1997, 69–94] that if there exists {C>0} such that \int_{B_{R}}(\varphi\sigma)^{2}\leq CR^{2} for every {R\geq 1}, then σ is necessarily constant. In this paper, we provide necessary and sufficient conditions on {0<\Psi\in C([1,\infty))} for which this result remains true if we replace {CR^{2}} by {\Psi(R)} in any dimension N. In the case of the convexity of Ψ for large {R>1} and {\Psi^{\prime}>0}, this condition is equivalent to \int_{1}^{\infty}\frac{1}{\Psi^{\prime}}=\infty.

scholarly journals On the existence of positive solutions for periodic parabolic sublinear problems

2003 ◽  
Vol 2003 (17) ◽  
pp. 975-984 ◽  
Author(s):  
T. Godoy ◽  
U. Kaufmann
Keyword(s):  
Bounded Domain ◽  
Positive Solutions ◽  
Wide Class ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Parabolic Problems ◽  
Smooth Bounded Domain ◽  
Carathéodory Functions ◽  
Existence Of Positive Solutions ◽  
Necessary And Sufficient

We give necessary and sufficient conditions for the existence of positive solutions for sublinear Dirichlet periodic parabolic problemsLu=g(x,t,u)inΩ×ℝ(whereΩ⊂ℝNis a smooth bounded domain) for a wide class of Carathéodory functionsg:Ω×ℝ×[0,∞)→ℝsatisfying some integrability and positivity conditions.

ON PROJECTIVELY RELATED WARPED PRODUCT FINSLER MANIFOLDS

2011 ◽  
Vol 08 (05) ◽  
pp. 953-967 ◽  
Author(s):  
M. M. REZAII ◽  
Y. ALIPOUR-FAKHRI
Keyword(s):  
Warped Product ◽  
Sufficient Conditions ◽  
Linear Connection ◽  
Necessary And Sufficient Conditions ◽  
Finsler Manifolds ◽  
Non Linear ◽  
Necessary And Sufficient

Let 𝔽1 = (M1,F1) and 𝔽2 = (M2,F2) be two Finsler manifolds and let M = M1 × M2 and S is a spray in M. Also 𝔽 = (M1 × f M2, F) is a warped product Finsler manifolds, such that the function f : M1 → ℝ+ is not constant. In this paper, we define a non-linear connection on warped product 𝔽, and finally, we have presented some necessary and sufficient conditions under which the spray manifold (M1 × M2, S) is projectively equivalent to the warped product Finsler manifolds (M1 × f M2, F).

scholarly journals Recurrence classification for a family of non-linear storage models

2018 ◽  
Vol 37 (2) ◽  
pp. 337-353
Author(s):  
Peter W. Glynn ◽  
Sanatan Rai ◽  
John E. Glynn
Keyword(s):  
Discrete Time ◽  
Power Law ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Storage Model ◽  
Positive Recurrence ◽  
Non Linear ◽  
Necessary And Sufficient ◽  
Recurrence Classification

RECURRENCE CLASSIFICATION FOR A FAMILY OF NON-LINEAR STORAGE MODELSNecessary and sufficient conditions for positive recurrence of a discrete-time non-linear storage model with power law dynamics arederived. In addition, necessary and sufficient conditions for finiteness of p-th stationary moments are obtained for this class of models.

scholarly journals An Extension of a Ger’s Result

2018 ◽  
Vol 32 (1) ◽  
pp. 263-274
Author(s):  
Dan Ştefan Marinescu ◽  
Mihai Monea
Keyword(s):  
Continuous Function ◽  
Functional Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Bregman Distance ◽  
International Conference ◽  
Necessary And Sufficient

Abstract The aim of this paper is to extend a result presented by Roman Ger during the 15th International Conference on Functional Equations and Inequalities. First, we present some necessary and sufficient conditions for a continuous function to be convex. We will use these to extend Ger’s result. Finally, we make some connections with other mathematical notions, as g-convex dominated function or Bregman distance.

scholarly journals On Ψ-instability of non-linear matrix Lyapunov systems

2009 ◽  
Vol 42 (4) ◽  
Author(s):  
M. S. N. Murty ◽  
G. S. Kumar ◽  
P. N. Lakshmi ◽  
D. Anjaneyulu
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Matrix ◽  
Non Linear ◽  
Necessary And Sufficient

AbstractWe prove necessary and sufficient conditions for Ψ-instability of trivial solutions of linear matrix Lyapunov systems and also sufficient conditions for Ψ-instability of trivial solutions of non-linear matrix Lyapunov systems.

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