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Title: Nonconvergent radial solutions of semilinear elliptic equations
Authors: Kwong, MK
Wong, SWH
Keywords: radial solutions
semilinear elliptic equations
Issue Date: 2010
Publisher: IOS Press
Source: Asymptotic analysis, 2010, v. 70, no. 1-2, p. 1-11 How to cite?
Journal: Asymptotic Analysis 
Abstract: For many known examples of semilinear elliptic equations Δu + f(u)=0 in ℝ N (N>1), a bounded radial solution u(r) converges to a constant as r→∞. Maier, in 1994, constructed, for N=2, an equation with a nonconvergent radial solution. Some necessary conditions for the existence of a nonconvergent solution were given by Maier, and later extended by Iaia. These conditions point out that, for N>2, equations with nonconvergent solutions are rather rare. A nonconvergent solution must oscillate between two constant values c 1<c 2 and f must vanish at either c 1 or c 2. In the neighborhood of one of these points, f must fluctuate wildly in an unusual way that excludes almost all common functions. In this paper, we give a further improvement of the above result with an alternative, simpler proof. The proof depends on an elementary, but nonobvious property of an initial value problem.
DOI: 10.3233/ASY-2010-0998
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