Increase of powers in the lower order term: a come back when the source term has a poor summability

We consider a nonlinear equation with a monotone operator acting in W01,p (Ω), a lower order term u|u|n−1, and a source term in Lm(Ω) with a poor summability m > 1 or even m = 1. When the power n increases and tends to ∞, we prove that the (conveniently defined) solution of this problem converges (in a convenient sense) to the solution of the variational inequality posed on the convex set {v ∈ W01,p (Ω) : |v(x)| ≤ 1}.