On the Infinitely Many Solutions of a Semilinear Elliptic Equation

Abstract

A dynamical systems approach is developed for studying the spherically symmetric solutions of $\Delta u + f(u) = 0$, where $f(u)$ grows like $| u |^\sigma u$ as $| u | \to \infty $ . Various scalings are introduced to elucidate the singular behavior near the center and at infinity. The solutions of interest appear as trajectories in a three-dimensional phase space with a different amount of oscillation around a certain invariant axis. Using this oscillation, solutions with a prescribed number of zeros can be found when ${{\sigma < 4} / {(n - 2)}}$.