Existence and non existence results for supercritical systems of Liouville-type equations on simply connected domains

We obtain a Pohozaev-type identity which yields a generalization to the systems case of the well known scalar non-existence threshold for Liouville-type mean field equa- tions on strictly starshaped domains. These newly derived non-existence results suggest that in principle solutions could be find in a region of parameters far away from the subcritical regime with respect to the vectorial Moser–Trudinger and Log-HLS inequalities found by Chipot, Shafrir and Wolansky. Indeed, we succeed in proving that the Dirichlet problem for cooperative Liouville systems admits solutions on “thin” simply connected domains in the supercritical regime. This is an improvement of the existence theory for cooperative Liou- ville systems since in that region solutions were known to exist only on multiply connected domains. Finally, combining spectral elliptic estimates and Orlicz-spaces techniques with a trick introduced by Wolansky we prove that these newly derived solutions are strict local minimizers of the Moser–Trudinger-type and free-energy functionals.