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.

• 35-02 - Research exposition (monographs, survey articles) pertaining to partial differential equations
• 35B09 - Positive solutions to PDEs
• 35B33 - Critical exponents in context of PDEs
• 35B38 - Critical points of functionals in context of PDEs (e.g., energy functionals)
• 35B45 - A priori estimates in context of PDEs
• 35B65 - Smoothness and regularity of solutions to PDEs
• 35J20 - Variational methods for second-order elliptic equations
• 35J50 - Variational methods for elliptic systems
• 35J61 - Semilinear elliptic equations
• 35R35 - Free boundary problems for PDEs
• 49Q05 - Minimal surfaces and optimization
• 58E05 - Abstract critical point theory (Morse theory, Lyusternik-Shnirel'man theory, etc.) in infinite-dimensional spaces
• 58J05 - Elliptic equations on manifolds, general theory