Lawrence C. Evans once wrote: “One important principle of mathematics is that extreme cases reveal interesting structure.” In this talk, we put this piece of mathematical wisdom to the test. First, we recall some known results on elliptic problems with power nonlinearity as the exponent tends to infinity. Then we consider the case of the Lane-Emden system and study the profile of least-energy solutions as one of the exponents tends to infinity. In this case, the limit profile can be characterized as a least-energy solution of a p-biharmonic nonlinear equation which can be studied with tools from nonsmooth analysis. These results were obtained in collaboration with Nicola Abatangelo (Università di Bologna) and Hugo Tavares (Universidade de Lisboa).