The existence and uniqueness of solutions to some nonlinear L φ -elliptic problems

In the present paper, we focus on the study of the nonlinear elliptic problem defined as {−div a(x,u,∇u)+g(x,u)=μ−div ϕ(x,u)in Ω,u=0on ∂Ω, where Ω is a bounded open domain of IRN (N≥2), the nonlinear term g(x,u) satisfies the sign condition, for the right-hand side μ=f−div(F) with f∈L1(Ω) and F∈(Eφ¯(Ω))N and ϕ(x,u) is a nonlinear term, satisfying some growth conditions. We prove the existence and uniqueness of renormalized solutions for this nonlinear elliptic problem in Musielak-Orlicz Sobolev spaces.