# Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity

@article{Cao2019ClassificationON,
title={Classification of nonnegative solutions to a bi-harmonic equation with Hartree type nonlinearity},
author={Daomin Cao and Wei Dai},
journal={Proceedings of the Royal Society of Edinburgh: Section A Mathematics},
year={2019},
volume={149},
pages={979 - 994}
}
• Published 1 August 2019
• Mathematics
• Proceedings of the Royal Society of Edinburgh: Section A Mathematics
Abstract In this paper, we are concerned with the following bi-harmonic equation with Hartree type nonlinearity $$\Delta ^2u = \left( {\displaystyle{1 \over { \vert x \vert ^8}}* \vert u \vert ^2} \right)u^\gamma ,\quad x\in {\open R}^d,$$where 0 < γ ⩽ 1 and d ⩾ 9. By applying the method of moving planes, we prove that nonnegative classical solutions u to (𝒫γ) are radially symmetric about some point x0 ∈ ℝd and derive the explicit form for u in the Ḣ2 critical case γ = 1. We also prove the non…

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