Logarithmic double phase problems with critical growth on the boundary

Abstract 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, {L} (u) = f (x, u) + |u|^p_*-2u on, 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 + (x) (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 |