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In this article, we prove an existence result of solutions to the obstacle problem associated with the equation of the typewhere A is an operator of Leray-Lions type acting from W 1, p(x) 0 ( ) into its dual W −1, p (x) ( ) and g(x, s, ξ) is a nonlinear term satisfying some growth condition,without the sign condition.

Abstract. The aim of this paper is to study the existence of solutions in the sense of distributions for a strongly nonlinear elliptic problem where the second term of the equation f is in W −1, − → p ′ ( · ) (Ω) which is the dual space of the anisotropic Sobolev(Ω) and later f will be in L 1 (Ω).

In this article, we study the problem (∂ b(x,u) ∕ ∂ t) - div a(x, t, u, ∇ u) + div φ(u) = f, in Ω × ]0, T], u = 0 on ∂ Ω × ]0,T[ b(x,u)(t=0) = b(x,u0). in Ω, in the framework of generalized Sobolev spaces, with b(x,u) unbounded function on u. The main contribution of our work is to prove the existence of renormalized solutions when the second term f belongs to L1(QT).

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