The main result of the paper provides the existence of a solution to a quasilinear inclusion problem with Dirichlet boundary condition which exhibits a term with full dependence on the solution and its gradient (convection term) and is driven by the degenerated p-Laplacian with weight. The multivalued term in the differential inclusion is in form of the generalized gradient of a locally Lipschitz function expressed through the primitive of a locally essentially bounded function, which makes the problem to be of a hemivariational inequality type. The novelty of our result is that we are able to simultaneously handle three major features: degenerated leading operator, convection term and discontinuous nonlinearity. Results of independent interest regard certain nonlinear operators associated to the differential inclusion.

Mathematics Subject Classification (2010): 35J87, 35J62, 35J70.

Keywords: Differential inclusion, hemivariational inequality, quasilinear elliptic equation, degenerated p-Laplacian with weight, Dirichlet problem, convection, pseudomonotone operator.