We consider the problem −∆u = χ {u>0} g (., u) + f (., u) in Ω, u = 0 on ∂Ω, u ≥ 0 in Ω, where Ω is a bounded domain in R n , f : Ω×[0, ∞) → R and g : Ω × (0, ∞) → [0, ∞) are Carathéodory functions, with g (x, .) nonnegative, nonincreasing, and singular at the origin. We establish sufficient conditions for the existence of a nonnegative weak solution 0 ≡ u ∈ H 1 0 (Ω) to the stated problem. We also provide conditions that guarantee that the found solution is positive a.e. in Ω. The problem with a parameter ∆u = χ {u>0} g (., u)+λf (., u) in Ω, u = 0 on ∂Ω, u ≥ 0 in Ω is also studied. For both problems, the special case when g (x, s) := a (x) s −α(x) , i.e., a singularity with variable exponent, is also considered.
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