I started in the following way:
$$1= \int_{_{\mathbb{R}^{n}}} g_{\lambda}^{p^*} (y) \, dy, \; \text{where } g_{\lambda} = (\lambda + |y|^{p'})^{\frac{p-n}{p}} \text{ and } p^* = \frac{pn}{p} $$
Then, $$1= \int_{_{\mathbb{R}^{n}}} g_{\lambda}^{p^*} (y) \, dy = \int_{_{\mathbb{R}^{n}}} (\lambda + |y|^{p'})^{-n} \, dy $$
And, by using the fact that $g_\lambda$ is a radial function and $n$ dimensional polar coordinates:
$$1=n \cdot w_n \cdot \int_{_{0}}^{\infty} (\lambda + \rho^{p'})^{-n} \cdot \rho^{n-1} \, d\rho$$ I could use that $n \cdot w_n = Area(S^{n-1}$), with radius $\rho$. I'm not sure how to continue form this point in order to find the form of $\lambda$.