A strong unique continuation property for weakly coupled elliptic systems

We establish the validity of a strong unique continuation property for weakly coupled elliptic systems, including competitive ones. Our proof exploits the system-structure of the problem and Carleman estimates. Then, we use our unique continuation theorems to show two nonexistence results. The first one states the nonexistence of nontrivial solutions to a weakly coupled elliptic system with a critical nonlinearity and Dirichlet boundary condition on starshaped domains, whereas the second one yields nonexistence of symmetric least energy solutions for a critical system in more general domains.