Boundedness of weak solutions to multivalued anisotropic double phase problems with logarithmic perturbation

In this paper we consider multivalued mixed boundary value problems driven by the variable exponent double phase operator with ω-logarithmic perturbation of the form $$ {cases}-divA(x,u,∇ u)(x,u,∇ u) & in Ω, \\ -{∂ u}{∂ νA} ∈ G(x,u) & on Γ₁, \\ u=0, & on Γ₂.{cases} $$ First, we prove new continuous and compact (trace) embedding results related to the considered Musielak-Orlicz Sobolev space $W^{1,HL}$. Based on these embedding results, we show the boundedness of solutions to the multivalued mixed boundary problem above by applying De Giorgi's method and localization arguments. Finally, we consider several special cases of the problem above and establish their boundedness results.