A blow-up approach for a priori bounds in semilinear planar elliptic systems: the Brezis-Merle critical case

We establish uniform a priori estimates for solutions of semilinear planar Hamiltonian elliptic systems in a ball with Dirichlet boundary conditions. We consider a broad class of coupled nonlinearities with asymptotic critical behaviour in the sense of Brezis–Merle. The approach we follow is based on a blow-up analysis combined with Liouville–type theorems and integral estimates. Our results extend the scalar theory of uniform a priori bounds to the Hamiltonian case, and solve an open problem in [de Figueiredo D.G., do Ó J.M., Ruf B., Adv. Nonlinear Stud. 6 (2006), no. 2]. We believe that this approach is new in this setting. As a consequence of our a priori estimates, we prove the existence of a positive solution by means of Fixed Point Index theory.