  In this paper we present a new bootstrap procedure for elliptic systems with
two unknown functions. Combining with the $L^p$-$L^q$-estimates, it yields the
optimal $L^\infty$-regularity conditions for the three well-known types of weak
solutions: $H_0^1$-solutions, $L^1$-solutions and $L^1_\delta$-solutions.
Thanks to the linear theory in $L^p_\delta(\Omega)$, it also yields the optimal
conditions for a priori estimates for $L^1_\delta$-solutions. Based on the a
priori estimates, we improve known existence theorems for some classes of
elliptic systems.
