Electric field for a radial power-law charge density ($\mathbf{E}(r)=\frac{A{r}^{p+1}}{{ϵ}_0(p+3)}\hat{\mathbf{r}}$)

Let the spherically symmetric volume charge density be $\rho (r)=A{r}^p$, with $p>-3$ so the origin contains finite charge. Integration gives $Q(r)=4\pi A{r}^{p+3}/(p+3)$. Gauss's law yields the radial electric field $\mathbf{E}=A{r}^{p+1}\hat{\mathbf{r}}/[{ϵ}_0(p+3)]$ for $r>0$. The field tends to zero at the centre if $p>-1$; finite enclosed charge alone does not guarantee a regular central field.