On a heat problem involving the perturbed Hardy–Sobolev operator

Abstract

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) = u₀ ≥ 0, u₀ ≢ 0, where L_λ* := −Δ − λ*/|x|², λ* := ¼(n − 2)², does not admit any 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 solutions for the above heat problem with the operator L_λ*q. In dimension n = 2, we have similar phenomena for the critical Hardy–Sobolev operator L* := −Δ − (1/4|x|²)(log R/|x|)⁻² for sufficiently large R.