Existence of renormalized solutions for nonlinear elliptic problems in weighted variable-exponent space with L^1-data

In this paper we study the existence of a renormalized solution for the nonlinear p(x)--elliptic problem in the Weighted--Variable--Exponent Soblev spaces, of the form: - div (a(x, u, ∇ u)) + H(x, u, ∇ u) = f ∈ Ω, where the right-hand side f belong to L1(Ω) and H(x, s, ξ) is the nonlinear term satisfying some growth condition, but no sign condition on s.