Let $a \geq 2 $.
How can we get, using Young's inequality, that $$ a^2 b^{2a-4} \leq \frac{1}{2} b^{2a-2}+ C a^{2a}$$
for some large enough C.
It seems like we are trying to apply it with $p=2$ and $q=2$ so we get
$$a^2 b^{2a-4} \leq \frac{1}{2} {(b^{2a-4})}^2+ \frac{1}{2} {(a^2)}^2 \leq \frac{1}{2} {(b^{2a-4})}^2+ C a^{2a}$$
since $a \geq 2$.
But I don't see how we get the first term to have the power $(2a-2)$?