Non‐convergent radial solution of a semilinear elliptic equation in $\mathbb{R}$ N

We examine the bounded radial solutions of Δu+f(u)=0 in $\mathbb{R}$ ^N with N>1. We show for many nonlinearities f that if lim _r→∞u(r)= does not exist then f≡0 on some nonempty open interval and N=2.