Isolated singularities of positive solutions of elliptic equations with weighted gradient term

Abstract

Let Ω ⊂ ℝN (N > 2) be a C2 bounded domain containing the origin 0. We study the behavior near 0 of positive solutions of equation (E) − Δu + |x|αup + |x|β|∇u|q = 0 in Ω ∖{0}, where α > −2, β > −1, p > 1, and q > 1. When 1 < p < (N + α)∕(N − 2) and 1 < q < (N + β)∕(N − 1), we provide a full classification of positive solutions of (E) vanishing on ∂Ω. On the contrary, when p ≥ (N + α)∕(N − 2) or (N + β)∕(N − 1) ≤ q ≤ 2 + β, we show that any isolated singularity at 0 is removable.