Abstract

We consider the problem of Ambrosetti-Prodi type{(Δ u + eu = s φ{symbol}1 + h (x), in  Ω,; u = 0, on  ∂ Ω,) where Ω is a bounded, smooth domain in R2, φ{symbol}1 is a positive first eigenfunction of the Laplacian under Dirichlet boundary conditions and h ∈ C0, α (over(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{symbol}1 as s → + ∞. © 2006 Elsevier Inc. All rights reserved.