  We analyse Hamiltonian-type systems of second-order elliptic PDE invariant
under a non-compact group and, consequently, involve a lack of compactness of
the Sobolev embedding. We show that the loss of compactness can be compensated
by using a concentration-compactness principle via weak profile decomposition
for bounded Palais-Smale sequences in Banach spaces. Our analysis to prove the
existence of ground states involves a reduction by the inversion method of the
system to a fourth-order equation combined with a variational principle of a
minimax nature. Among other results, including regularity and a Pohozaev-type
identity, we also prove the non-existence of weak solutions for a class of
Lane-Emden systems.
