Extremal Positive Solutions of Semilinear Schrödinger Equations

1983 ◽  
Vol 26 (2) ◽  
pp. 171-178 ◽  
Author(s):  
C. A. Swanson
Keyword(s):  
Differential Equation ◽  
Positive Solutions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Exterior Domains ◽  
Hölder Continuous ◽  
Semilinear Differential Equation ◽  
Necessary And Sufficient

AbstractNecessary and sufficient conditions are proved for the existence of maximal and minimal positive solutions of the semilinear differential equation Δu = -ƒ(x, u) in exterior domains of Euclidean n-space. The hypotheses are that ƒ(x, u) is nonnegative and Hölder continuous in both variables, and bounded above and below by ugi(| x |, u), i = 1, 2, respectively, where each gi(r, u) is monotone in u for each r > 0.

Asymptotic Theory of Singular Semilinear Elliptic Equations

1984 ◽  
Vol 27 (2) ◽  
pp. 223-232 ◽  
Author(s):  
Takaŝi Kusano ◽  
Charles A. Swanson
Keyword(s):  
Positive Solutions ◽  
Elliptic Equations ◽  
Asymptotic Theory ◽  
Gordon Equation ◽  
Sufficient Conditions ◽  
Semilinear Elliptic Equation ◽  
Hölder Continuous ◽  
Semilinear Elliptic ◽  
Klein Gordon ◽  
Necessary And Sufficient

AbstractNecessary and sufficient conditions are found for the existence of two positive solutions of the semilinear elliptic equation Δu + q(|x|)u = f(x, u) in an exterior domain Ω⊂ℝn, n ≥ 1, where q, f are real-valued and locally Hölder continuous, and f(x, u) is nonincreasing in u for each fixed x∈Ω. An example is the singular stationary Klein-Gordon equation Δu — k2u = p(x)u-λ where k and λ are positive constants. In this case NASC are given for the existence of two positive solutions ui(x) in some exterior subdomain of Ω such that both |x|m exp[(-l)i-1k|x|]ui(x) are bounded and bounded away from zero in this subdomain, m = (n —1)/2, i = 1, 2.

scholarly journals Distributional properties of solutions of dVt = Vt-dUt + dLt with Lévy noise

2011 ◽  
Vol 43 (3) ◽  
pp. 688-711 ◽  
Author(s):  
Anita Diana Behme
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Autocorrelation Function ◽  
Sufficient Conditions ◽  
Stationary Solutions ◽  
Necessary And Sufficient Conditions ◽  
Tail Behavior ◽  
Absolutely Continuous ◽  
Distributional Properties ◽  
Necessary And Sufficient

For a given bivariate Lévy process (Ut, Lt)t≥0, distributional properties of the stationary solutions of the stochastic differential equation dVt = Vt-dUt + dLt are analysed. In particular, the expectation and autocorrelation function are obtained in terms of the process (U, L) and in several cases of interest the tail behavior is described. In the case where U has jumps of size −1, necessary and sufficient conditions for the law of the solutions to be (absolutely) continuous are given.

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