| Abstract: |
| I will discuss some recent results concerning uniform a priori bounds for positive solutions of elliptic equations and semilinear Hamiltonian elliptic systems involving exponential-type non-linearities in dimension two. These topics are presented in joint works in collaboration with Laura Baldelli (Karlsruhe Institute of Technology), Luca Battaglia (Universit\`{a} degli Studi Roma Tre), Giulio Romani (Universit\`{a} degli Studi di Udine) and Pierre-Damien Thizy (Universit\`{e} Claude Bernard Lyon 1). In the case of systems, 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. As a consequence of our a priori estimates, we prove the existence of a positive solution by means of fixed point index theory. |
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