Decay estimates and symmetry of finite energy solutions to elliptic systems in \mathbb{R}^n

2019 ◽  
Vol 68 (3) ◽  
pp. 663-696
Author(s):  
Jerome Vetois
Keyword(s):  
Elliptic Systems ◽  
Finite Energy ◽  
Decay Estimates

scholarly journals SYMMETRY AND EXISTENCE OF SOLUTIONS OF SEMI-LINEAR ELLIPTIC SYSTEMS IN HYPERBOLIC SPACE

2014 ◽  
Vol 57 (3) ◽  
pp. 519-541
Author(s):  
HAIYANG HE
Keyword(s):  
Positive Solution ◽  
Hyperbolic Space ◽  
Existence Of Solutions ◽  
Elliptic Systems ◽  
Ground State Solution ◽  
Decay Estimates ◽  
Symmetry Properties ◽  
Formula Group ◽  
Solution Pair ◽  
Image Position

Abstract(0.1) \begin{equation}\label{eq:0.1} \left\{ \begin{array}{ll} \displaystyle -\Delta_{\mathbb{H}^{N}}u=|v|^{p-1}v x, \\ \displaystyle -\Delta_{\mathbb{H}^{N}}v=|u|^{q-1}u, \\ \end{array} \right. \end{equation} in the whole Hyperbolic space ℍN. We establish decay estimates and symmetry properties of positive solutions. Unlike the corresponding problem in Euclidean space ℝN, we prove that there is a positive solution pair (u, v) ∈ H1(ℍN) × H1(ℍN) of problem (0.1), moreover a ground state solution is obtained. Furthermore, we also prove that the above problem has a radial positive solution.

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.

Positive solutions of elliptic systems with bounded nonlinearities

1988 ◽  
Vol 108 (3-4) ◽  
pp. 321-332 ◽  
Author(s):  
Ezzat S. Noussair ◽  
Charles A. Swanson
Keyword(s):  
Positive Solutions ◽  
Weak Coupling ◽  
Existence Of Solutions ◽  
Elliptic Systems ◽  
Decay Estimates ◽  
Exterior Domains ◽  
Differential Systems ◽  
Asymptotic Decay ◽  
Semilinear Elliptic ◽  
Exponential Decay Law

SynopsisSemilinear elliptic partial differential systems of second order with weak coupling are considered in exterior domains Ω ⊆ ℝN, N≧3. Conditions on the nonlinearities are given which guarantee the existence of solutions u with positive components in Ω such that u|∂Ω = 0 and u(x)→0 uniformly as |x|→∞. Asymptotic decay estimates for the solutions are established, including an exponential decay law under extra hypotheses.

Essential Spectrum and Exponential Decay Estimates of Solutions of Elliptic Systems of Partial Differential Equations. Applications to Schrödinger and Dirac Operators

2008 ◽  
Vol 15 (2) ◽  
pp. 333-351
Author(s):  
Vladimir S. Rabinovich ◽  
Steffen Roch
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Essential Spectrum ◽  
Exponential Decay ◽  
Elliptic Systems ◽  
Dirac Operators ◽  
Decay Estimates ◽  
Partial Differential ◽  
Limit Operators ◽  
Estimates Of Solutions

Abstract The main aim of the paper is to study relations between the location of the essential spectrum and exponential decay estimates of the eigenfunctions of systems of elliptic partial differential equations. To solve this problem, we apply the limit operators method. Applications are given to the problem of exponential decay behavior of eigenfunctions of Schrödinger and Dirac operators.

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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