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

Abstract

We study the vectorial stationary Schrödinger equation $-\Delta u+aU+bu=F$, with a saturated nonlinearity $U=u/\left|u\right|$ 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 $F\left(x\right)$ is not vanishing for large values of $\left|x\right|$. 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 Schrödinger–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.