Let's say we have a PDE, for example the Laplace equation: $$ \Delta u = f. $$ Usually, to solve such a thing, one finds its variational formulation, and solves it in some Sobolev space.
My question is, is it possible to avoid the weak formulation, and find solutions directly in Hölder spaces? Maybe via some fixed-point argument?
A good book to start is Gilbard-Trudinger chapter 6. Also take a look in Brezis book. After Theorem 9.33 (Schauder), he makes a comment where he gives a plenty of references about the subject.