Optimal conditions for -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 L p – L q -estimates, it yields the optimal L ∞ -regularity conditions for the three well known types of weak solutions: H 0 1 -solutions, L 1 -solutions and L δ 1 -solutions. Thanks to the linear theory in L δ p ( Ω ) , it also yields the optimal conditions for a priori estimates for L δ 1 -solutions. Based on the a priori estimates, we improve known existence theorems for some classes of elliptic systems.