# 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.