Elliptic boundary value problems with nonsmooth potential and mixed boundary conditions

The goal of this article is to establish the existence of at least one solution for a boundary value problem governed by a quasilinear elliptic operator and two multivalued functions given by Clarke's generalized gradient of some locally Lipschitz functionals. We divide the boundary of our domain into two open measurable parts (1) and (2) and we impose a nonhomogeneous Neumann boundary condition on (1), while on (2) we impose a Dirichlet boundary condition. This kind of problems have been treated in the past by various authors. However, in all the work we are aware of either a Neumann or a Dirichlet boundary condition imposed on the entire boundary. Another main point of interest of this article is that our problem depends on a real parameter >0 and we are able to prove the existence of solutions for any (0,+).