Abstract: For elliptic systems with discontinuous nonlinearities, we study the existence of strong solutions whose values are points of continuity with respect to the state variables for almost all values of the spatial variable. Such solutions are said to be semiregular. An upper-and-lower-solution principle is established for the existence of semiregular solutions to elliptic systems with discontinuous nonlinearities. This principle is used to prove theorems on the existence of semiregular solutions of elliptic systems with discontinuous nonlinearities, in particular, nontrivial solutions of problems with a parameter. Examples of classes of nonlinearities with separated variables satisfying the conditions of our theorems are given.