fa.functional analysis – Reference request: PDE of the form $(Delta – |u|^2)f = F(u)$

I am interested in equations of the form
$$(Delta -|u|^2)f = F(u)$$
where $F$ depends on $u$ and preferably on its derivative, too. $u$ is supposed to be given and $f$ the unknown. More precisely I am interested in how to obtain a priori bounds on $f$, i.e.
$$ |f|_X leq |(Delta-|u|^2)^{-1} F(u)|_Y$$
Of course, if we were to replace $(Delta-|u|^2)^{-1}$ by, say, $Delta^{-1}$ we could use something like Hardy-Littlewood-Sobolev to obtain bounds which is not possible if you take the term $|u|^2$ into account. If you can help me out with references on this kind of equation I would be very grateful.