  We prove the existence of unique smooth solutions to the quaternionic
Monge-Amp\`{e}re equation for $(n-1)$-quaternionic plurisubharmonic functions
on a hyperK\"{a}hler manifold and thus obtain solutions for the quaternionic
form type equation. We derive $C^0$ estimate by establishing a Cherrier-type
inequality as in Tosatti and Weinkove [22]. By adopting the approach of Dinew
and Sroka [9] to our context, we obtain $C^1$ and $C^2$ estimates without
assuming the flatness of underlying hyperK\"{a}hler metric comparing to
previous results [14].
