Vector quasi-hemivariational inequalities and discontinuous elliptic systems

We develop an existence theory for hemivariational inequalities in vector-valued function spaces which involve pseudomonotone operators. The obtained abstract result is used to study quasilinear elliptic systems whose lower order coupling vector field depends discontinuously upon the solution vector. We provide conditions that allow the identification of regions of existence of solutions for such systems, so called trapping regions.