Regularity issues for semilinear PDE-s (a narrative approach)

Occasionally, solutions of semilinear equations have better (local) regularity properties than the linear ones if the equation is independent of space (and time) variables. The simplest example, treated by the current author, was that the solutions of $\Delta u=f(u)$, with the mere assumption that $f'\geq-C$, have bounded second derivatives. In this paper, some aspects of semilinear problems are discussed, with the hope to provoke a study of this type of problems from an optimal regularity point of view. It is noteworthy that the above result has so far been undisclosed for linear second order operators, with Hölder coefficients. Also, the regularity of level sets of solutions as well as related quasilinear problems are discussed. Several seemingly plausible open problems that might be worthwhile are proposed.