In the previous chapter, the elliptic problems considered were linear. However, the modelling of many phenomena gives rise to elliptic problems that contain nonlinearities and that deserve the attention of mathematicians. Semi-linear PDEs are the closest to linear PDEs, in the sense that higher order derivatives appear as linear terms, with coefficients that are functions of the space variables; non-linearities can appear in the terms of the lower order derivatives and of the right-hand side.