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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 the dual space of the anisotropic Sobolev and later f will be in L1. key words: Anisotropic Sobolev spaces, variable exponent, strongly nonlinear elliptic equa- tions, boundary value problems, solution in the sense of distributions.