Oscillation Criteria for a Class of Perturbed Schrödinger Equations

1982 ◽  
Vol 25 (1) ◽  
pp. 71-77 ◽  
Author(s):  
Takaŝi Kusano ◽  
Manabu Naito
Keyword(s):  
Elliptic Equation ◽  
Euclidean Space ◽  
Exterior Domain ◽  
Continuous Functions ◽  
Oscillatory Behavior ◽  
Dimensional Euclidean Space ◽  
Order Elliptic Equation ◽  
Second Order Elliptic Equation ◽  
Image Position ◽  
The Laplace Operator

We are concerned with the oscillatory behavior of the second order elliptic equation1where Δ is the Laplace operator inn-dimensional Euclidean spaceRn,Eis an exterior domain inRn, andc:E × R → Randf:E → Rare continuous functions.A functionv : E − Ris called oscillatory inEifv(x) has arbitrarily large zeros, that is, the set {x∈E:v(x) = 0} is unbounded. For brevity, we say that equation (1) is oscillatory inEif every solutionu∈C2(E) of (1) is oscillatory inE.

scholarly journals ON THE OSCILLATION OF AN ELLIPTIC EQUATION OF FOURTH ORDER

1996 ◽  
Vol 27 (2) ◽  
pp. 151-159
Author(s):  
BHAGAT SINGH
Keyword(s):  
Elliptic Equation ◽  
Laplace Operator ◽  
Fourth Order ◽  
Exterior Domain ◽  
Oscillatory Behavior ◽  
All Solutions ◽  
The Laplace Operator

The elliptic equation \[\Delta^2 u(|x|)+g(|x|)u(|x|)=f(|x|)\] is studied for its oscillatory behavior. $\Delta$ is the Laplace operator. Sufficient condi­ tions have been found to ensure that all solutions of this equation continuable in some exterior domain $\Omega=\{x=(x_1, x_2, x_3):|x|>A\}$ where $|x|=(\sum_{i=1}^3 x_i^2)^{1/2}$ are oscillatory.  

scholarly journals EXISTENCE OF POSITIVE EVANESCENT SOLUTIONS TO SOME QUASILINEAR ELLIPTIC EQUATIONS

2008 ◽  
Vol 78 (1) ◽  
pp. 157-162 ◽  
Author(s):  
OCTAVIAN G. MUSTAFA
Keyword(s):  
Elliptic Equation ◽  
Positive Solution ◽  
Elliptic Equations ◽  
Exterior Domain ◽  
Quasilinear Elliptic Equations ◽  
Image Position ◽  
Quasilinear Elliptic

AbstractWe establish that the elliptic equation defined in an exterior domain of ℝn, n≥3, has a positive solution which decays to 0 as $\vert x\vert \rightarrow +\infty $ under quite general assumptions upon f and g.

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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