  Using the concept of strong necessary conditions (CSNC), we derive Bogomolny
equations and BPS bounds for two modifications of the gauged baby BPS Skyrme
model: the nonminimal coupling to the gauge field and k-deformed model. In
particular, we study, how the Bogomolny equations and the equation for the
potential, reflect these two modifications. In both examples, the CSNC method
shows to be a very useful tool.
