Oscillatory Radial Solutions of Semilinear Elliptic Equations

Abstract

We study the oscillatory behavior of radial solutions of the nonlinear partial differential equation Δu + f(u) + g(|x|, u) = 0 inRn, where f and g are continuous restoring functions, uf(u) > 0 and ug(|x|, u) > 0 for u ≠ 0. We assume that for fixedq limu → 0(|f(u)|/|u|q) = B > 0, for 1 < q < n/(n − 2), and, additionally, that 2F(u) ≥ (1 − 2/n)uf(u) when n/(n − 2) ≤ q < (n + 2)/(n − 2), where F(u) = ∫u0f(s)ds. We give conditions that guarantee that the solution oscillates infinitely and tends to zero asr → ∞. Finally, we give bounds for the amplitude of the oscillations and show that the period of the oscillations increases asr → ∞