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Asymptotic Theory of Singular Semilinear Elliptic Equations

Published online by Cambridge University Press:  20 November 2018

Takaŝi Kusano  and
Charles A. Swanson
Takaŝi Kusano
Affiliation:
Department of Mathematics, Hiroshima University, Hiroshima, Japan
Charles A. Swanson
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver B.C., CanadaV6T 1Y4
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Abstract

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Necessary 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.

Keywords

35B0535J6034C1134EXX
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 27 , Issue 2 , 01 June 1984 , pp. 223 - 232
Copyright
Copyright © Canadian Mathematical Society 1984

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