  In this paper, which corresponds to an updated version of the author's
Habilitation lecture in Mathematics, we do an overview of several topics in
elliptic problems. We review some old and new results regarding the Lane-Emden
equation, both under Dirichlet and Neumann boundary conditions, then focus on
sign-changing solutions for Lane-Emden systems. We also survey some results
regarding fully nontrivial solutions to gradient elliptic systems with mixed
cooperative and competitive interactions. We conclude by exhibiting results on
optimal partition problems, with cost functions either related to Dirichlet
eigenvalues or to the Yamabe equation. Several open problems are referred along
the text.
