Optimal conditions for L∞-regularity and a priori estimates for semilinear elliptic systems

In this paper, we present a bootstrap procedure for semilinear elliptic systems with n (⩾3) components. Combining with the Lp–Lq-estimates, it yields the optimal L∞-regularity conditions for the three well known types of weak solutions: -solutions, L1-solutions and -solutions. Thanks to the linear theory in , it also yields the optimal conditions for a priori estimates for -solutions. Based on the a priori estimates, we improve known existence theorems for some classes of elliptic systems.