Oscillatory radial solutions of semilinear elliptic equations (Q1359584)

The authors study oscillatory solutions of \(-\left(u''+{n-1\over r} u'\right)= f(u)\) on \([0,\infty)\). The function \(f\) is sign-preserving and subcritical: \[ 0<uf(u)\leq{2n\over n-2}\int^u_0 f(s)ds \text{ for } u\neq 0. \] Their first theorem states that if \(\lim_{u\to 0}|f(u)|/|u|^q= B\neq 0\) for \(q\in\left(1, {n+2\over n-2}\right)\) and if \(u\) is a solution, then \(\lim_{r\to\infty} u(r)=0\) and \(u\) will oscillate infinitely. The next two theorems give, in the case that \(f(u)=(b|u|^{q-1}+|u|^{p-1})u\), with \(1<q<p\leq {n+2\over n-2}\), the decay rate of the extrema and a bound from below for the `period'.