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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).

Our aim in this paper is to study the existence of renormalized solution for a class of nonlinear p(x)-Laplace problems with Neumann nonhomogeneous boundary conditions and diuse Radon measure data which does not charge the sets of zero p(.)-capacity

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