= Solution
For an outward radial <electric field>, <Gauss's law> gives the enclosed charge
$$
Q(r)=4\pi\epsilon_0r^2 E_r=4\pi\epsilon_0\alpha r e^{-\lambda r}.
$$
Taking the enclosing sphere to infinity gives \b[$Q_{\rm total}=0$]. A nonzero local field is compatible with zero net charge. Indeed, for $r>0$,
$$
\rho(r)=\epsilon_0\nabla\cdot\mathbf E
=\frac{\epsilon_0\alpha e^{-\lambda r}}{r^2}(1-\lambda r).
$$
Positive inner charge is balanced by negative outer charge. There is no point charge at the origin, since $\lim_{r\to0}4\pi\epsilon_0r^2 E_r=0$. The field's magnitude does not specify its inward/outward sign, but reversing a globally radial direction leaves the zero total unchanged. This is <net charge from the asymptotic radial electric field>.
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