Stable weak solutions of weighted nonlinear elliptic equations

This paper deals with the weighted nonlinear elliptic equation -div(|x|^α∇_u) = |x|γe^u in Ω u = 0 on ∂Ω where α, γ ∈ ℝ satisfy N + α > 2 and γ -α > -2, and the domain Ω ⊂ ℝ^N(N ≥ 2) is bounded or not. Moreover, when α ≠ 0, we prove that, for N + α > 2, γ-α ≤-2, the above equation admits no weak solution. We also study Liouville type results for the equation in ℝ^N.