In 1994, while working at the Paris Dauphine University, Lions received the International Mathematical Union's prestigious Fields Medal.

His first published article, in 1977, was a contribution to the vast literature on convergence of certain iterative algorithms to fixed points of a given nonexpansive self-map of a closed convex subset of Hilbert space.

[13] With Bertrand Mercier, Lions proposed a "forward-backward splitting algorithm" for finding a zero of the sum of two maximal monotone operators.

[L82b] With Henri Berestycki and Lambertus Peletier, Lions used standard ODE shooting methods to directly study the existence of rotationally symmetric solutions.

[BL83a] By adapting the critical point methods of Felix Browder, Paul Rabinowitz, and others, Berestycki and Lions also demonstrated the existence of infinitely many (not always positive) radially symmetric solutions to the PDE.

[BL83b] Maria Esteban and Lions investigated the nonexistence of solutions in a number of unbounded domains with Dirichlet boundary data.

[BL83a] They showed that such identities can be effectively used with Nachman Aronszajn's unique continuation theorem to obtain the triviality of solutions under some general conditions.

[16] Significant "a priori" estimates for solutions were found by Lions in collaboration with Djairo Guedes de Figueiredo and Roger Nussbaum.

[FLN82] In more general settings, Lions introduced the "concentration-compactness principle", which characterizes when minimizing sequences of functionals may fail to subsequentially converge.

[DLM91] In the physical sense, such results, known as velocity-averaging lemmas, correspond to the fact that macroscopic observables have greater smoothness than their microscopic rules directly indicate.

[DL89a] DiPerna and Lions' results on the transport equation were later extended by Luigi Ambrosio to the setting of bounded variation, and by Alessio Figalli to the context of stochastic processes.

[CEL84] Using a min-max quantity, Lions and Jean-Michel Lasry considered mollification of functions on Hilbert space which preserve analytic phenomena.

Using such techniques, Crandall and Lions extended their analysis of Hamilton-Jacobi equations to the infinite-dimensional case, proving a comparison principle and a corresponding uniqueness theorem.