scholarly journals Semilinear Second Order Elliptic Oscillation

1979 ◽  
Vol 22 (2) ◽  
pp. 139-157 ◽  
Author(s):  
C. A. Swanson
Keyword(s):  
Partial Differential Equations ◽  
Euclidean Space ◽  
Recent Progress ◽  
Unbounded Domains ◽  
Elliptic Partial Differential Equations ◽  
Second Order ◽  
Dimensional Euclidean Space ◽  
Semilinear Elliptic ◽  
Oscillation Problem ◽  
Image Position

These pages summarize recent progress on the oscillation problem for semilinear elliptic partial differential equations of the form(1)in unbounded domains Ω in n-dimensional Euclidean space Rn. Our attention is restricted to the second order symmetric equation (1), and completeness is not attempted; the emphasis is on results obtained in the last five years.

Radial Entire Solutions of Even Order Semilinear Elliptic Equations

1988 ◽  
Vol 40 (6) ◽  
pp. 1281-1300 ◽  
Author(s):  
Takaŝi Kusano ◽  
Manabu Naito ◽  
Charles A. Swanson
Keyword(s):  
Partial Differential Equations ◽  
Continuous Function ◽  
Positive Integer ◽  
Positive Constant ◽  
Elliptic Equations ◽  
Elliptic Partial Differential Equations ◽  
Entire Solutions ◽  
Semilinear Elliptic ◽  
Even Order ◽  
Image Position

Semilinear elliptic partial differential equations of the type1will be considered throughout real Euclidean N-space, where m ≧ 2 is a positive integer, Δ denotes the N-dimensional Laplacian, and f is a real-valued continuous function in [0, ∞) × (0, ∞). Detailed hypotheses on the structure of f are listed in Section 3.Our objective is to prove the existence of radially symmetric positive entire solutions u(x) of (1) which are asymptotic to positive constant multiples of |x|2m−2i as |x| → ∞ for every i = 1,…, m, N ≧ 2i + 1.

Properties of Harmonic Functions of Three Real Variables Given by Bergman-Whittaker Operators

1963 ◽  
Vol 15 ◽  
pp. 157-168 ◽  
Author(s):  
Josephine Mitchell
Keyword(s):  
Euclidean Space ◽  
Harmonic Functions ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Rectifiable Curve ◽  
Image Position

Let be a closed rectifiable curve, not going through the origin, which bounds a domain Ω in the complex ζ-plane. Let X = (x, y, z) be a point in three-dimensional euclidean space E3 and setThe Bergman-Whittaker operator defined by

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