Logarithmic double phase problems with critical growth on the boundary

In this paper, we study logarithmic double phase problems with critical growth on the boundary of the form aligned -div L(u)=-|u|ᵖ⁻²u in Ω , L(u)· ν = f(x,u)+ |u|p_*-2u on ∂ Ω , aligned - div L ( u ) = - | u | p - 2 u in Ω , L ( u ) · ν = f ( x , u ) + | u | p ∗ - 2 u on ∂ Ω , where div L div L stands for the logarithmic double phase operator given by $${aligned} div ( |∇ u|ᵖ⁻² ∇ u + μ (x) [ log (e + |∇ u|) + |∇ u|/q(e + |∇ u|) ] |∇ u|q-2 ∇ u ) , {aligned}$$ div | ∇ u | p - 2 ∇ u + μ ( x ) log ( e + | ∇ u | ) + | ∇ u | q ( e + | ∇ u | ) | ∇ u |