On the compactness of the support of solitary waves of the complex saturated nonlinear Schrödinger equation and related problems

We study the vectorial stationary Schr & ouml;dinger equation - + + = , with a saturated nonlinearity = /| | and with some complex coefficients ( , ) is an element of C 2 . Besides the existence and uniqueness of solutions for the Dirichlet and Neumann problems, we prove the compactness of the support of the solution, under suitable conditions on ( , ) and even when the source in the right hand side () is not vanishing for large values of ||. The proof of the compactness of the support uses a local energy method, given the impossibility of applying the maximum principle. We also consider the associate Schr & ouml;dinger-Poisson system when coupling with a simple magnetic field. Among other consequences, our results give a rigorous proof of the existence of "solitons with compact support"claimed, without any proof, by several previous authors.