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 positive entire solutions for weakly coupled semilinear elliptic systems

1992 ◽  
Vol 120 (1-2) ◽  
pp. 79-91 ◽  
Author(s):  
Yasuhiro Furusho
Keyword(s):  
Sufficient Conditions ◽  
Elliptic Systems ◽  
Real Constant ◽  
Entire Solutions ◽  
Semilinear Elliptic Systems ◽  
Semilinear Elliptic ◽  
Weakly Coupled ◽  
Non Linear ◽  
Image Position

SynopsisWeakly coupled semilinear elliptic systems of the formare considered in RN, N≧2, where k = 1, 2, …, M, u = (u1, …, uM) and λ is a real constant. The aim of this paper is to give sufficient conditions for (*) to have entire solutions whose components are positive in RN and converge to non-negative constants as |x| tends to ∞. For this purpose a new supersolution-subsolution method is developed for the system (*) without any hypotheses on the monotonicity of the non-linear terms fk with respect to u.

Article

1999 ◽  
Vol 77 (11) ◽  
pp. 1810-1812 ◽  
Author(s):  
Alex D Bain
Keyword(s):  
Spin System ◽  
Coupled System ◽  
Coupled Systems ◽  
Physical Reason ◽  
Strongly Coupled ◽  
System Combination ◽  
First Order ◽  
Weakly Coupled ◽  
Quantum Transitions ◽  
Weakly Coupled Systems

Strongly coupled spin systems provide many curious and interesting effects in NMR spectra, one of which is the presence of unexpected (from a first-order viewpoint) lines. A physical reason is given for the presence of these combination lines. The X part of the spectrum of an ABX spin system is analysed as an example. For an ABX system, it is well known that the AB nuclei give a spectrum consisting of two AB-type spectra, corresponding to the two orientations of the X nucleus. It can also be shown that the X part of the spectrum corresponds to the X nucleus undergoing a transition in the presence of an AB-like spin system. For weakly coupled systems, the four observed lines correspond to the four different orientations of the A and B nuclei. For a strongly coupled system, two additional lines may appear, the combination lines. The resulting six lines correspond to the four spin orientations, plus the two zero-quantum transitions. It is shown that these six lines are such that there is no net excitation of the AB-like spin system associated with the X transitions. There is no AB coherence created directly by a pulse applied to X. AB coherence is created as the system evolves, and this is responsible for many of the curious effects. This is shown to be true for all spin sub-systems, which are weakly coupled to a strongly coupled sub-system.Key words: NMR, strong coupling, second-order spectra, ABX spin system, combination lines, spectral analysis.

scholarly journals Finite energy solutions to inhomogeneous nonlinear elliptic equations with sub-natural growth terms

2020 ◽  
Vol 13 (1) ◽  
pp. 53-74 ◽  
Author(s):  
Adisak Seesanea ◽  
Igor E. Verbitsky
Keyword(s):  
Sufficient Conditions ◽  
Nonlinear Elliptic Equations ◽  
Classical Case ◽  
Finite Energy ◽  
Natural Growth ◽  
Nonlinear Elliptic ◽  
Borel Measures ◽  
Inhomogeneous Problems ◽  
Necessary And Sufficient ◽  
Quasilinear Elliptic

AbstractWe obtain necessary and sufficient conditions for the existence of a positive finite energy solution to the inhomogeneous quasilinear elliptic equation-\Delta_{p}u=\sigma u^{q}+\mu\quad\text{on }\mathbb{R}^{n}in the sub-natural growth case {0<q<p-1}, where {\Delta_{p}} ({1<p<\infty}) is the p-Laplacian, and σ, μ are positive Borel measures on {\mathbb{R}^{n}}. Uniqueness of such a solution is established as well. Similar inhomogeneous problems in the sublinear case {0<q<1} are treated for the fractional Laplace operator {(-\Delta)^{\alpha}} in place of {-\Delta_{p}}, on {\mathbb{R}^{n}} for {0<\alpha<\frac{n}{2}}, and on an arbitrary domain {\Omega\subset\mathbb{R}^{n}} with positive Green’s function in the classical case {\alpha=1}.

Transport in weakly coupled systems

1973 ◽  
Vol 59 (6) ◽  
pp. 3235-3243
Author(s):  
Gary R. Dowling ◽  
H. T. Davis
Keyword(s):  
Coupled Systems ◽  
Weakly Coupled ◽  
Weakly Coupled Systems

Oscillation of Semilinear Elliptic Inequalities by Riccati Transformations

1980 ◽  
Vol 32 (4) ◽  
pp. 908-923 ◽  
Author(s):  
E. S. Noussair ◽  
C. A. Swanson
Keyword(s):  
Exterior Domain ◽  
Nonlinear Term ◽  
Sufficient Conditions ◽  
Type Inequality ◽  
Second Derivative ◽  
Semilinear Elliptic ◽  
Generalized Riccati Transformation ◽  
The Matrix ◽  
Elliptic Inequality ◽  
Elliptic Inequalities

A generalized Riccati transformation will be utilized to derive a Riccati-type inequality (3) associated with a semilinear elliptic inequality yL(y; x) ≦ 0 possessing a positive solution y in an exterior domain in Euclidean n-space. On the basis of (3), general sufficient conditions for the elliptic inequality to be oscillatory are developed in § 3. The matrix of coefficients of the second derivative terms in L(y;x) (i.e. (Aij) in (1)) is not restricted in any way beyond the usual ellipticity hypothesis (iv) below, and thereby one of the difficulties mentioned in [9] and inherent in the method there is resolved. Furthermore, the nonlinear term B﹛x, y) in (1) is not required to be one-signed.

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