 Solution

The functions are nonnegative and normalized to have integral one. On the disk $|x|\le (2n{)}^{-1/4}$, Bernoulli inequality gives $(1-|x{|}^4{)}^n\ge 1-n|x{|}^4\ge 1/2$. ConsequentlyFor any fixed $0<a<1$, the mass outside radius $a$ is at mostThis is an approximate identity. For any test function,Let $n\to \infty$ and then $a\downarrow 0$ to conclude $\boxed{g_n\to {\delta }_0\text{ in }{\mathcal{D}}^{\prime }({\mathbb{R}}^2)}$.

Polar coordinates also give the exact normalization through the beta function: