Existence, Uniqueness & Bifurcations · Superchallenge problem

For $p > 0$ let $f(y) = y\,(-\ln y)^p$ for $0 < y < e^{-1}$, $f(0) = 0$, and extend $f$ oddly ($f(-y) = -f(y)$). The IVP $y' = f(y)$, $y(0) = 0$ always has the solution $y \equiv 0$. Find the largest $p$ for which this is the ONLY solution.