Closing the gap in the purely elliptic generalized Davey-Stewartson system

In this note we improve the results presented previously on global existence and global nonexistence for the solutions of the purely elliptic generalized Davey-Stewartson system. These results left a gap in the parameter range where neither a global existence result nor a global nonexistence result could be established. Here we are able to show that when the coupling parameter is negative there is no gap. Moreover, in the case where the coupling parameter is positive we reduce the size of the gap.