A system of two elliptic equations with discontinuous nonlinearities and homogeneous Dirichlet boundary conditions is studied. Existence theorems for strong and semiregular solutions are deduced using a variational method. A strong solution is called semiregular if the set on which the values of the solution are points of discontinuity of the nonlinearity with respect to the phase variable has measure zero. Classes of nonlinearities are distinguished for which the assumptions of the theorems established here hold. The variational approach in this paper is based on the concept of a quasipotential operator, by contrast with the traditional approach, which uses the generalized Clark gradient. Bibliography: 22 titles.