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Radial entire solutions to even order semilinear elliptic equations in the plane

Published online by Cambridge University Press:  14 November 2011

Takaŝi Kusano ,
Manabu Naito  and
Charles A. Swanson
Takaŝi Kusano
Affiliation:
Department of Mathematics, Faculty of Science, Hiroshima University, 1-1-89 Higashi-Senda, Naka-Ku, Hiroshima 730, Japan
Manabu Naito
Affiliation:
Department of Mathematics, Faculty of Education, Tokushima University, 1-1 Minami-Josanjima, Tokushima 770, Japan
Charles A. Swanson
Affiliation:
Department of Mathematics, University of British Columbia, 121-1984 Mathematics Road, Vancouver, B.C., CanadaV6T 1Y4
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Synopsis

Semilinear elliptic equations of the form

are considered, where Δm is the m-th iterate of the two-dimensional Laplacian Δ, p(t) is continuous in [0, ∞), and f(u is continuous and positive either in (0, ∞) or in ℝ.

Our main objective is to present conditions on p and f which imply the existence of radial entire solutions to (*), that is, those functions of class C2m(ℝ2) which depend only on |x| and satisfy (*) pointwise in ℝ2.

First, necessary and sufficient conditions are established for equation (*), with p(t) > 0 in [0, ∞), to possess infinitely many positive radial entire solutions which are asymptotic to positive constant multiples of |x|2m−2 log |x as |x| → ∞. Secondly, it is shown that, in the case p(t < 0, in [ 0, ∞) and f(u) > 0 is nondecreasing in ℝ, equation (*) always has eventually negative radial entire solutions, all of which decrease at least as fast as negative constant multiples of |x|2m−2 log |x| as |x| → ∞. Our results seem to be new even when specialised to the prototypes

where γ is a constant.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 105 , Issue 1 , 1987 , pp. 275 - 287
Copyright
Copyright © Royal Society of Edinburgh 1987

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