I am trying to understand the well-posedness of some semilinear elliptic problems, such as:

$$-\Delta u +F(x,u)=0, \ \Omega$$ $$u=f \in C^{2,\alpha}, \ \partial \Omega, $$ or $$-\Delta u +F(x,\nabla u)=0, \ \Omega$$ $$u=f \in C^{2,\alpha}, \ \partial \Omega, $$

where the $F$ has nice properties, such as being positive and $\alpha-$Hölder in both variables and $C^{1,\alpha}$ in the second one. For the continuity of the map $f \rightarrow u_{f}$ we understand that we employ Hölder norms.

I have never studied these topics deeply before. I guess the general strategy is to employ the Inverse Function Theorem (its version in Banach space). So I would proceed by studying the Frechet derivative, checking that it is invertible... But I don't seem to be able to prove a Global inversion result, nor the continuous dependence on the boundary data. Also, the calculus of variations, which is the only thing I've ever studied having something to do with non-linear equations, does not seem to help here.

When I have more time, in three or more months, I intend to study thoroughly the book by Gilbarg and Trudinger, but right now I need these very particular results for a more concrete work, and I would like to understand these result in relatively short time.

Therefore, if someone could sketch the proof of the well-posedness, or give references of where to find these concrete problems, I would be really grateful.


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