Standing waves of nonlinear Schrödinger systems with all attractive forces

Since the pioneering work of Lin and Wei on nonlinear Schrödinger systems of $n$ components with interaction forces $a_{ij}$ between the $i$-th and $j$-th components for $1⩽i,j⩽n$, there have been numerous further developments in many directions. However, even in the simplest case where all interaction forces are attractive, (which corresponds to $a_{ij}⩾0$ for every $1⩽i,j⩽n$), a systematic approach to finding a solution with all positive components is still not known. In the special case $n=2$, a very systematic approach has been developed by Ambrosetti and Colorado. In this paper, we review the fundamental works by Lin–Wei and Ambrosetti–Colorado and further developments.