This may be too general a question so please let me know if I need to make it more specific.
I am a first year graduate student in PDEs, and as such have not had much exposure to non-linear PDEs. I am starting to look at research papers, in many of which the PDE at hand is non-linear, and the 'linearization' of such a problem has been considered without much justification as to why this is useful. For instance, on page 4 here, after the authors have established that we wish to show the existence of a solution to a fully non-linear PDE, they suddenly shift to talking about the linearized problem and I have little idea of
1) How this linearized problem is derived, and
2) How the existence of a solution to the linearized problem relates to the existence of a solution to the original problem.
From what I gather in the short document here, we can only linearize 'about a known solution', but what if we have no known solution? Even if we do have a known solution, question 2 above still stands.
I guess more specifically I am asking for either a reference on the linearization of non-linear PDEs (i.e. how it is done, why it is useful), or if possible an answer explaining this concept, either in relation to the paper I linked or more generally. I have done two courses in elliptic PDE theory (covering chapters 5 and 6 in Evans in the first course, and going quite far beyond that in the second course) and have still never encountered linearization - hopefully that gives some indication of my background. Thank you.