  We are concerned with the two-power nonlinear Schr\"odinger-type equations
with non-local terms. We consider the framework of Sobolev-Lorentz spaces which
contain singular functions with infinite-energy. Our results include global
existence, scattering and decay properties in this singular setting with
fractional regularity index. Solutions can be physically realized because they
have finite local $L^2$-mass. Moreover, we analyze the asymptotic stability of
solutions and, although the equation has no scaling, show the existence of a
class of solutions asymptotically self-similar w.r.t. the scaling of the
single-power NLS-equation. Our results extend and complement those of [F.
Weissler, ADE 2001], particularly because we are working in the larger setting
of Sobolev-weak-$L^p$ spaces and considering non-local terms. The two
nonlinearities of power-type and the generality of the non-local terms allow us
to cover in a unified way a large number of dispersive equations and systems.
