On the behavior at infinity of solutions of elliptic systems with a finite energy integral

1987 ◽  
Vol 99 (1) ◽  
pp. 75-89 ◽  
Author(s):  
V. A. Kondratiev ◽  
O. A. Oleinik
Keyword(s):  
Elliptic Systems ◽  
Finite Energy ◽  
Energy Integral

scholarly journals Connected sets of positive solutions of elliptic systems in exterior domains

2019 ◽  
Vol 191 (4) ◽  
pp. 761-778
Author(s):  
Aleksandra Orpel
Keyword(s):  
Elliptic System ◽  
Positive Solutions ◽  
Exterior Domain ◽  
Main Tool ◽  
Elliptic Systems ◽  
Finite Energy ◽  
Exterior Domains ◽  
Asymptotic Decay ◽  
Semilinear Equations ◽  
Connected Sets

Abstract The existence of infinitely many connected sets of positive solutions for a certain elliptic system is investigated in this paper. We consider semilinear equations with perturbed Laplace operators described in an exterior domain. We show that each of these solutions $$\mathbf {u}=( u_{1},u_{2})$$u=(u1,u2) has the minimal asymptotic decay, namely $$ u_{i}(x)=O(||x||^{2-n})$$ui(x)=O(||x||2-n) as $$||x||\rightarrow \infty ,$$||x||→∞,$$i=1,2,$$i=1,2, and finite energy in a neighborhood of infinity. Our main tool is the sub and super-solutions theorem which is based on the Sattinger’s iteration procedure. We do not need any growth assumptions concerning nonlinearities.

Asymptotics for Semilinear Elliptic Systems

1991 ◽  
Vol 34 (4) ◽  
pp. 514-519 ◽  
Author(s):  
Ezzat S. Noussair ◽  
Charles A. Swanson
Keyword(s):  
Exterior Domain ◽  
Sufficient Conditions ◽  
Elliptic Systems ◽  
Finite Energy ◽  
Elliptic Partial Differential Equations ◽  
Coupled Systems ◽  
Semilinear Elliptic ◽  
Weakly Coupled ◽  
Weakly Coupled Systems ◽  
Necessary And Sufficient

AbstractA class of weakly coupled systems of semilinear elliptic partial differential equations is considered in an exterior domain in ℝN, N > 3. Necessary and sufficient conditions are given for the existence of a positive solution (componentwise) with the asymptotic decay u(x) = O(|x|2-N) as |x| —> ∞. Additional results concern the existence and structure of positive solutions u with finite energy in a neighbourhood of infinity.

EXISTENCE OF FINITE ENERGY SOLUTIONS FOR ELLIPTIC SYSTEMS WITH L1-VALUED NONLINEARITIES

2008 ◽  
Vol 18 (05) ◽  
pp. 669-687 ◽  
Author(s):  
LUCIO BOCCARDO ◽  
LUIGI ORSINA ◽  
ALESSIO PORRETTA
Keyword(s):  
Elliptic System ◽  
Sobolev Spaces ◽  
Open Subset ◽  
Existence Of Solutions ◽  
Elliptic Systems ◽  
Finite Energy ◽  
Carathéodory Functions ◽  
Bounded Open Subset ◽  
Existence And Regularity Results ◽  
Regularity Results

In this paper, we are going to study the following elliptic system: [Formula: see text] where Ω is a bounded open subset of ℝN, a(x, s) and b(x, s) are positive and coercive Carathéodory functions, and f ∈ LM(Ω). The main purpose of this paper is to prove existence and regularity results with an improved regularity of the function z in the class of Sobolev spaces, and the existence of solutions (u, z) both with finite energy.

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