A generalization of the Hopf-Cole transformation for stationary Mean Field Games systems

In this note we propose a transformation which decouples stationary Mean Field Games systems with superlinear Hamiltonians of the form |p|^r, and turns the Hamilton-Jacobi-Bellman equation into a quasi-linear equation involving the r-Laplace operator. Such a transformation requires an assumption on solutions of the system, which is satisfied for example in space dimension one or if solutions are radial

Separation of Coupled Systems of Schrodinger Equations by Darboux transformations

Darboux transformations in one independent variable have found numerous applications in various field of mathematics and physics. In this paper we show that the extension of these transformations to two dimensions can be used to decouple systems of Schrodinger equations and provide explicit representation for three classes of such systems. We show also that there is an elegant relationship between these transformations and analytic complex matrix functions.Comment: 14 page

Nonzero positive solutions of a multi-parameter elliptic system with functional BCs

We prove, by topological methods, new results on the existence of nonzero positive weak solutions for a class of multi-parameter second order elliptic systems subject to functional boundary conditions. The setting is fairly general and covers the case of multi-point, integral and nonlinear boundary conditions. We also present a non-existence result. We provide some examples to illustrate the applicability our theoretical results.Comment: 10 page