Find the best constant $\|u\|_{L^r}\leq C\|\nabla u\|_{L^p}^a\|u\|_{L^q}^{1-a}$.

Question

Can someone give me a reference for computing the optimal constant $C$ (independent of $u$) in the inequality $$\|u\|_{L^r(\mathbb R^d)}\leq C\|\nabla u\|_{L^p(\mathbb R^d)}^a\|u\|_{L^q(\mathbb R^d)}^{1-a},$$ where $u\in\mathcal C^1_c(\mathbb R^d)$ radial and $$\frac{1}{r}=\frac{a}{p}+\frac{1-a}{q}$$ with $p,q\geq 1$ and $a\in [0,1]$ and $d\geq 1$.