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

We consider the problem of Ambrosetti–Prodi type{Δu+eu=sϕ1+h(x)in Ω,u=0on ∂Ω, where Ω   is a bounded, smooth domain in R2R2, ϕ1ϕ1 is a positive first eigenfunction of the Laplacian under Dirichlet boundary conditions and h∈C0,α(Ω¯). We prove that given k⩾1k⩾1 this problem has at least k   solutions for all sufficiently large s>0s>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 ϕ1ϕ1 as s→+∞s→+∞.