In this paper we investigate the weak solvability of differential inclusions with mixed boundary conditions of the type \( (P): \quad \left\{ \textstyle\begin{array}{l@{\quad }l} -{\mathrm{div}}\left(\frac{\phi '(|\nabla u|)}{|\nabla u|} \nabla u\right)+ \frac{\phi '(|u|)}{|u|}u\in \partial _{C} f(x,u), & \text{ in }\Omega , \\ u=0, & \text{ on }\Gamma _{1} \\ \frac{\phi '(|\nabla u|)}{|\nabla u|} \frac{\partial u}{\partial \nu } \in \partial _{C} g(x,u),& \text{ on }\Gamma _{2}, \end{array}\displaystyle \right. \) where $\Omega \subset \mathbb{R}^{N}$ is a bounded domain with Lipschitz boundary $\partial \Omega =\bar{\Gamma }_{1}\cup \bar{\Gamma }_{2}$ , $\Gamma _{1}\cap \Gamma _{2}=\varnothing $ , $\phi :[0,\infty )\to [0,\infty )$ is an $N$ -function of $C^{1}$ class, $f(x,t)$ and $g(x,t)$ are measurable w.r.t. the first variable and locally Lipschitz w.r.t. the second variable such that their corresponding Clarke subdifferential satisfy an Orlicz-type growth condition. Using nonsmooth critical point theory we are able to prove existence and multiplicity results provided the function $\phi $ either completely dominates or it is completely dominated by both functions controlling the growth of $\partial _{C} f(x,\cdot )$ and $\partial _{C} g(x,\cdot )$ , respectively. An important feature of the paper is that we allow ${\mathrm{meas}}(\Gamma _{i})=0$ , $i\in \{1,2\}$ , thus obtaining in the limiting case either a pure Dirichlet or Neumann-type boundary condition.
