Abstract: We study the homogeneous Dirichlet problem for a second-order elliptic equation with anonlinearity discontinuous in the state variable in the resonance case. A class of resonanceproblems that does not overlap with the previously investigated class of strongly resonanceproblems is singled out. Using the variational method, we establish a theorem on the existence ofat least three nontrivial solutions of the problem under study (the zero is its solution). In thiscase, at least two nontrivial solutions are semiregular; i.e., the values of such solutions fall on thediscontinuities of the nonlinearity only on a set of measure zero. We give an example of anonlinearity satisfying the assumptions of this theorem. A sufficient semiregularity condition isobtained for a nonlinearity with subcritical growth at infinity, a case which is of separate interest.Applications of the theorem to problems with a parameter are considered. The existence ofnontrivial (including semiregular) solutions of the problem with a parameter for an ellipticequation with a discontinuous nonlinearity for all positive values of the parameter is established.