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In this paper we consider the nonlinear Dirichlet problem: A(u)=T in Ω0 on ∂Ω, where the right hand side belongs to a “large” Sobolev space W1.q(Ω), 1 < q < 2, for some q, and A is a strongly monotone operator acting from W1.20(Ω) into W1.2(Ω), defined by: A(υ)=-div (a(x,▽υ)). If the differential operator A is linear, the existence of solutions in a “large” Sobolev space has been obtained by N. G. Meyers (see [8]), using a duality method and a regularity theorem. In the case where in the previous differential problem the right hand side is zero, but with non-zero boundary conditions, the existence of solutions can be found in [7]. Other nonlinear boundary value problems with weak solutions in “large” Sobolev spaces are studied in [2] and [1], where the right hand side is a bounded measure.