On critical logarithmic double phase problems with locally defifined perturbation

This paper deals with critical logarithmic double phase problems of the form −div????(u)=g(x,u)+|u|p∗−2u in Ω, u=0 on ∂Ω, where div????is the logarithmic double phase operator defifined by div(︃|∇u|p−2∇u+μ(x)(︃log(e+|∇u|)+|∇u| q(e+|∇u|))︃|∇u|q−2∇u)︃, eis Euler’s number, Ω⊂RN, N≥2, is a bounded domain with Lipschitz boundary ∂Ω, 1