Multiple blowing-up solutions to critical elliptic systems in bounded domains

Abstract

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.