The density of any point inside the sphere is

$\rho ={\rho }_0\left(1-\frac{r^2}{R^2}\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}0<r\le R$

weight = buoyant force
mg = B

$mg={\rho }_{L}Vg$
$\int \rho \left(4\pi r^{2}dr\right)g={\rho }_L\frac{4}{3}\pi R^{3}g$
$\underset{0}{\overset{R}{\int }}{\rho }_0\left(1-\frac{r^2}{R^2}\right)4\pi r^{2}dr={\rho }_L\frac{4}{3}\pi R^3$
$\underset{0}{\overset{R}{\int }}{\rho }_{0}4\pi \left(r^2-\frac{r^4}{R^2}\right)dr={\rho }_{0}4\pi {\left(\frac{r^3}{3}-\frac{r^5}{5R^2}\right)}_0^R={\rho }_L\frac{4}{3}\pi R^3$
$\frac{2}{5}{\rho }_0={\rho }_L$