Asymptotic behavior of solutions of semilinear elliptic equations near an isolated singularity

We consider the semilinear elliptic equation Δ u = h (u) in Ω {set minus} {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, C_N) (that is, lim_{u → ∞} h (λ u) / h (u) = λ^q, for every λ > 0), where C_N 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.