On a heat problem involving the perturbed Hardy-Sobolev operator

Let Ω be a bounded domain in ℝⁿ n ≥ 3 and 0 ∈ Ω. It is known that the heat problem ∂u/∂t + L _λ*u = 0 in Ω × (0, ∞), u(x,0) = u ₀ ≥ 0, u₀ ≢ 0, where L _λ*:= -Δ - λ*/|x|², λ*:= 1/4(n - 2)², does not admit any H₀ ¹ solutions for any t > 0. In this paper we consider the perturbation operator L_λ*q:= -Δ - λ*q(x)/|x|² for some suitable bounded positive weight function q and determine the border line between the existence and non-existence of positive H₀¹ solutions for the above heat problem with the operator L_λ*q. In dimension n = 2, we have similar phenomena for the critical Hardy-Sopolev operator L*:= -Δ - (1/4|x|²)(log R/|x|)^-2 for sufficiently large R.