  In this paper, we search for the BPS skyrmions in some BPS submodels of the
generalized Skyrme model in five-dimensional spacetime using the BPS Lagrangian
method. We focus on the static solutions of the Bogomolny's equations and their
corresponding energies with topological charge $B>0$ is an integer. We consider
two main cases based on the symmetry of the effective Lagrangian of the BPS
submodels, i.e. the spherically symmetric and non-spherically symmetric cases.
For the spherically symmetric case, we find two BPS submodels. The first BPS
submodels consist of a potential term and a term proportional to the square of
the topological current. The second BPS submodels consist of only the Skyrme
term. The second BPS submodel has BPS skyrmions with the same topological
charge $B>1$, but with different energies, that we shall call "topological
degenerate" BPS skyrmions. It also has the usual BPS skyrmions with equal
energies, if the topological charge is a prime number. Another interesting
feature of the BPS skyrmions, with $B>1$, in this BPS submodel, is that these
BPS skyrmions have non-zero pressures in the angular direction. For the
non-spherically symmetric case, there is only one BPS submodel, which is
similar to the first BPS submodel in the spherically symmetric case. We find
that the BPS skyrmions depend on a constant $k$ and for a particular value of
$k$ we obtain the BPS skyrmions of the first BPS submodel in the spherically
symmetric case. The total static energy and the topological charge of these BPS
skyrmions also depend on this constant. We also show that all the results found
in this paper satisfy the full field equations of motions of the corresponding
BPS submodels.
