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We prove the existence of a solution for a strongly nonlinear degenerated problem associated to the equation
Au + g(x,u,∇u) = f,
where A is a Leray–Lions operator from the weighted Sobolev space W01,p(Ω, w) into its dual W −1,p'(Ω, w*). While g(x,s,ξ) is a nonlinear term having natural growth with respect to ξ and no growth with respect to s, it satisfies a sign condition on s, i.e., g(x,s,ξ) · s ≥ 0 for every s∈ℝ. The right-hand side f belongs to L1(Ω).