On the compactness of the support of solitary waves of the complex saturated nonlinear Schr{ö}dinger equation and related problems

We study the vectorial stationary Schr{ö}dinger equation -$\mathrm{\Delta }$u + a U + b u = F, with a saturated nonlinearity U = u/|u| and with some complex coefficients (a, b) $\in$ 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 (a, b) and even when the source in the right hand side F (x) is not vanishing for large values of |x|. 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{ö}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.