Weak semiregular solutions to the Dirichlet problem for quasilinear elliptic equations in divergence form with discontinuous weak nonlinearities

In a bounded domain of an $N$-dimensional space we study the homogeneous Dirichlet problem for a quasilinear elliptic equation in divergence form with a discontinuous weak nonlinearity of power growth at infinity. Using a variational method based on the concept of quasipotential operator we obtain a theorem on the existence of a weak semiregular solution to the problem under study. The semiregularity of a solution means that, almost everywhere in the domain in which the boundary value problem is considered, its values are continuity points of the weak nonlinearity with respect to the phase variable. Next, a positive parameter is introduced into the equation as a multiplier of the weak nonlinearity, and the question of the existence of nontrivial weak semiregular solutions to the resulting boundary value problem is studied. In this case the existence of a trivial solution for all values of the parameter is assumed. A theorem on the existence of a nontrivial weak semiregular solution for sufficiently large values of the parameter is established.