A Divergence-type Identity in a Punctured Domain and its Application to a Singular Polyharmonic Problem

abstract

The divergence identity for punctured domain B₁(0)\ {0}

$$ \int_{\partial B_1(0)}\frac{\partial u}{\partial r}\,{\rm d}\sigma= \int_{B_1(0)}\Delta u\ \hbox{d}x \quad \mbox{with $r=|x|$},$$

suggest a viewpoint on describing the behavior of a function u∈C²(B₁(0)\{0}) near the origin. This is useful especially on describing the singular behavior of solutions of polyharmonic equations. In this paper we mainly show that the solution u of the equation

satisfies the identity that, letting v _i =(−Δ)ⁱu

provided there exist s₀>0 and t₀ ≥ 0 such that f(x,t)≥ c|x|^−σt^q for 0<|x|<s₀ and t ≥ t₀ with σ ≥ n−q(n−2p) and q>1.