Optimal conditions for a priori estimates for semilinear elliptic systems with two components

In this paper, we present a new bootstrap procedure for elliptic systems with two unknown functions. When combined with the LpLp-LqLq-estimates, it yields the optimal L∞L∞-regularity conditions for the three well-known types of weak solutions: H01-solutions, L1L1-solutions and Lδ1-solutions. Thanks to the linear theory in Lδp(Ω), it also yields the optimal conditions for a priori estimates for Lδ1-solutions. Based on the a priori estimates, we improve known existence theorems for some classes of elliptic systems.