Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities

This paper is concerned with the asymptotic behavior of solutions of the nonlinear elliptic eigenvalue problem

−Δu=λf(u), u>0 in Ω, u=0 on ∂Ω,

for λ↓0, where Ω is a bounded domain in R² and f(u) is an exponentially dominated nonlinear function. Under appropriate assumptions, we show that as λ↓0, {Σ}={λf_Ωe^udx} for solutions {u} accumulate to 0, 8πm, or +∞, where m is a positive integer. Moreover, according to these cases, the {u} converge to 0 uniformly, blow up exactly at m points, or everywhere in Ω.