Nonconvergent radial solutions of semilinear elliptic equations

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 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.