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 L-p-L-q-estimates, it yields the optimal L-infinity-regularity conditions for the three well-known types of weak solutions: H-0(1)-solutions, L-1-solutions and L-delta(1)-solutions. Thanks to the linear theory in L-delta(p)(Omega), it also yields the optimal conditions for a priori estimates for L-delta(1)-solutions. Based on the a priori estimates, we improve known existence theorems for some classes of elliptic systems. (C) 2009 Elsevier Ltd. All rights reserved.