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

We consider the problem of Ambrosetti–Prodi type where Ω is a bounded, smooth domain in , 1 is a positive first eigenfunction of the Laplacian under Dirichlet boundary conditions and . 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 1 as s→+∞.