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In this paper, we study a general class of nonlinear anisotrpic elliptic problems associated with the differential inclusion β(u) - div(a(x, D(u) + F(u)) ∋ f in Ω, where f ∈ L∞(Ω). A vector field a(. , .) is a Caratheodory function. Using trunction techniques and the generalized monotonicity method in the functional spaces we prove the existence of renormalized solutions for L∞-data.

In this paper, we study in the anisotropic Sobolev spaces with variable exponent W01, p(.)(Ω), the existence result of unilateral problem associated to the equations of the form, -∑i=1N ∂i ai(x, u, ∇ u) = g, where the right hand side f belongs to L1(Ω).

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.