On the finite energy classes of quaternionic plurisubharmonic functions

We investigate the finite $p$-energy classes $E_p$ of quaternionic plurisubharmonic functions of Cegrell type. We also construct an example to show that the optimal constant in the energy estimate is strictly bigger than $1$ for $p>0$, $p\neq 1$. This leads us to the fact that we can not use the variational method to solve the quaternionic Monge-Amp\`ere equation for the classes $E_p$ when $p>0$, $p\neq 1$.