  In this work we consider the initial-value problem associated with a coupled
system of generalized Korteweg-de Vries equations. We present a relationship
between the best constant for a Gagliardo-Nirenberg type inequality and a
criterion for the existence of global solutions in the energy space. We prove
that such a constant is directly related to the existence problem of
solitary-wave solutions with minimal mass, the so called ground state
solutions. To guarantee the existence of ground states we use a variational
method.
