The two-dimensional Lazer-McKenna conjecture for an exponential nonlinearity

We consider the problem of Ambrosetti-Prodi type [GRAPHICS] where Q is a bounded, smooth domain in R-2, phi(1) is a positive first eigenfunction of the Laplacian under Dirichlet boundary conditions and h is an element of C-0,C-alpha(Omega). 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 phi(1) as s +infinity