Existence and Multiplicity Results for Differential Inclusions with Orlicz Growth and Mixed Boundary Conditions

Abstract In this paper we investigate the weak solvability of differential inclusions with mixed boundary conditions of the type $$ (P): \quad \left\{ \textstyle\begin{array}{l@{\quad }l} -{\mathrm{div}}\left(\frac{\phi '(|\nabla u|)}{|\nabla u|} \nabla u\right)+ \frac{\phi '(|u|)}{|u|}u\in \partial _{C} f(x,u), & \text{ in }\Omega , \\ u=0, & \text{ on }\Gamma _{1} \\ \frac{\phi '(|\nabla u|)}{|\nabla u|} \frac{\partial u}{\partial \nu } \in \partial _{C} g(x,u),& \text{ on }\Gamma _{2}, \end{array}\displaystyle \right. $$ ( P ) : { − div ( ϕ ′ ( | ∇ u | ) | ∇ u | ∇ u ) + ϕ ′ ( | u | ) | u | u ∈ ∂ C f ( x , u ) , in Ω , u = 0 , on Γ 1 ϕ ′ ( | ∇ u | ) | ∇ u | ∂ u ∂ ν ∈ ∂ C g ( x , u ) , on Γ 2 , where $\Omega \subset \mathbb{R}^{N}$ Ω ⊂ R N is a bounded domain with Lipschitz boundary $\partial \Omega =\bar{\Gamma }_{1}\cup \bar{\Gamma }_{2}$ ∂ Ω = Γ ¯ 1 ∪