Nonconvergent radial solutions of semilinear elliptic equations

Article type: Research Article

Authors: Kwong, Man Kam | Wong, Solomon Wai-Him

Affiliations: Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong, SAR, China. E-mails: mankwong@polyu.edu.hk, mawhwong@inet.polyu.edu.hk

Abstract: For many known examples of semilinear elliptic equations Δu+f(u)=0 in RN (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 c1<c2 and f must vanish at either c1 or c2. 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.

Keywords: radial solutions, semilinear elliptic equations

DOI: 10.3233/ASY-2010-0998

Journal: Asymptotic Analysis, vol. 70, no. 1-2, pp. 1-11, 2010