Abstract
The systems considered have the form ▪ in Ω, ▪ on ∂Ω, where ▪ is a bounded domain, A is a matrix of second order elliptic operators, and γ is a real parameter. For simplicity the results are stated for a single equation, but the range of validity for systems is discussed. The first type of a priori estimates give lower bounds for sup ▪ in terms of γ and |Ω| when ▪ is superlinear, upper bounds for sup ▪ when ▪ is sublinear, and lower bounds for γ when ▪ has linear growth. The second type of estimates generalize to systems results of Brezis and Turner and P. L. Lions for single equations; they give upper bounds for sup ▪ in the superlinear case. Those estimates require A to be diagonal. None of the results require a variational structure for the system.