## Abstract

Let Ω be a bounded domain in ℝ^{n} 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, u_{0} ≢ 0, where L _{λ*}:= -Δ - λ*/|x|^{2}, λ*:= 1/4(n - 2)^{2}, does not admit any H_{0} ^{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 H_{0}^{1} 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.

Original language | English |
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Pages (from-to) | 683-693 |

Number of pages | 11 |

Journal | Proceedings of the Royal Society of Edinburgh Section A: Mathematics |

Volume | 134 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2004 |