I am studying the following Schrödinger equation: $$(-\Delta + w)f = 0$$ which represents a quantum state with zero energy. Here $w$ and $f$ are defined on $\mathbb{R}^{3}$. For simplicity, let us assume $w$ is nonnegative, radially symmetric $w(x) = w(|x|) = w(r)$, $r \in (0,\infty)$ and with compact support.

Assume the differential equation above holds in the distribution sense and suppose the solution $f$ is $f \in H^{1}(\mathbb{R}^{3})$. I am actually interested in radially symmetric solutions $f = f(r)$. My question is: how regular the derivative of $f$ can be near the origin? I know I can be the case that $\frac{df}{dr}$ goes to infinity as $r \to 0^{+}$, but is there any condition on, say, $r^{2}\frac{df}{dr}$ or some power $r^{n}\frac{df}{dr}$, $n \ge 1$ in which this product goes to zero as $r \to 0^{+}$?

Edit: The two important cases for me are: $w \in C^{\infty}$ and $w \in L^{3}(\mathbb{R}^{3})$.