  Normalized ground state solutions (NGSS) of Schrodinger equations (SE) have
attracted the attention of many research groups during the last decades. This
is essentially due to their relevance in many fields in physics and
engineering, where the stable and most attractive solutions happen to be the
normalized ones. For a single (SE), recent developments lead to the
establishment of existence and non-existence results for a wide range of
natural nonlinearities in (SE) in the sub-critical, critical and super-critical
regimes. However for systems of (SE), there are still many interesting open
questions for basic nonlinearities. It certainly requires innovative ideas to
shed some light to treat these complex situations. So far, only a very few
specific nonlinearities have been addressed. Unlike the single (SE), the
corresponding strict sub-additivity inequality is challenging and an improved
concentration-compactness theorem is critical to treat (NGSS) of systems of
(SE). The aim of this paper is to establish the existence of (NGSS) for a large
class of nonlinearities. This class includes many relevant pure-power type
nonlinearities and can be easily extended to Hartree type nonlinearities (and a
combination of both). The presence of the fractional Laplacian adds a
considerable difficulty to rule out the dichotomy. We were able to overcome
this challenge and establish very general assumptions ensuring the strict
subadditivity of the constrained energy functional. We believe that our
approach will open the door to many unresolved problems.
