Asymptotic Behavior of Solutions of Semilinear Elliptic Equations Near an Isolated Singularity

Abstract

We consider the semilinear elliptic equation Δu=h(u) in Ω {0}, where Ω is an open subset of ℝ^N (N≥2) containing the origin and h is locally Lipschitz continuous on [0,∞), positive in (0,∞). We give a complete classification of isolated singularities of positive solutions when h varies regularly at infinity of index q ∈ (1,CN)(that is, limu→∞h(λu)/h(u)=λ^q, for every λ>0), where CN denotes either N/(N-2) if N≥3 or ∞ if N=2. Our result extends a well-known theorem of Véron for the case h(u)=u^q.