  We construct families of blowing-up solutions to elliptic systems on smooth
bounded domains in the Euclidean space, which are variants of the critical
Lane-Emden system and analogous to the Brezis-Nirenberg problem. We find a
function which governs blowing-up points and rates, observing that it reflects
the strong nonlinear characteristic of the system. By using it, we also prove
that a single blowing-up solution exists in general domains, and construct
examples of contractible domains where multiple blowing-up solutions are
allowed to exist. We believe that a variety of new ideas and arguments
developed here will help to analyze blowing-up phenomena in related
Hamiltonian-type systems.
