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.
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2$\begingroup$ I'm a bit confused: seeing as $u$ is given, isn't this just a linear PDE? $\endgroup$– Leo MoosCommented Oct 30, 2020 at 10:33
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1$\begingroup$ Looks very linear to me indeed! $\endgroup$– leo monsaingeonCommented Oct 30, 2020 at 12:15
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$\begingroup$ @LeoMoos I know it is linear but I need to know how to estimate the RHS (say $|u|^2$ is known to be in some $L^p$ for $1\leq p \leq \infty$ in terms of the LHS. The usual Hardy-Littlewood-Sobolev inequality doesn't apply here, right? $\endgroup$– Jakob MöllerCommented Nov 2, 2020 at 13:07
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$\begingroup$ @Jakob: Have you checked the books of Han-Lin or Gilbarg-Trudinger for an answer to your question? Off the top of my head, the answer to your question might depend on whether $u \in L^p$ for some $p > n$. Could you be more precise about the specific estimate you are looking for? I also think there might be a typo in the inequality you state, should it be $\lVert f \rVert \leq C \lVert F(u) \rVert$? $\endgroup$– Leo MoosCommented Nov 3, 2020 at 12:16
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