We consider the problem of Ambrosetti–Prodi type { Δ u + e u = s ϕ 1 + h ( x ) in Ω , u = 0 on ∂ Ω , where Ω is a bounded, smooth domain in R 2 , ϕ 1 is a positive first eigenfunction of the Laplacian under Dirichlet boundary conditions and h ∈ C 0 , α ( Ω ¯ ) . We prove that given k ⩾ 1 this problem has at least k solutions for all sufficiently large s > 0 , which answers affirmatively a conjecture by Lazer and McKenna [A.C. Lazer, P.J. McKenna, On the number of solutions of a nonlinear Dirichlet problem, J. Math. Anal. Appl. 84 (1981) 282–294] for this case. The solutions found exhibit multiple concentration behavior around maxima of ϕ 1 as s → + ∞ .