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

Article type: Research Article

Authors: Nagasaki, Ken'ichi | Suzuki, Takashi

Affiliations: Department of Mathematics, Chiba Institute of Technology, Narashino, Japan 275 | Department of Mathematics, Faculty of Science, Tokyo Metropolitan University, Tokyo, Japan 158

Abstract: 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}={λfΩeudx} 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.

DOI: 10.3233/ASY-1990-3205

Journal: Asymptotic Analysis, vol. 3, no. 2, pp. 173-188, 1990