A Divergence-Type Identity in a Punctured Domain and Its Application to a Singular Polyharmonic Problem

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Abstract

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

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 vi = (−Δ)iu

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

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