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

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  27 December 2018

Daomin Cao  and
Wei Dai
Daomin Cao
Affiliation:
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510405, People's Republic of China and Institute of Applied Mathematics, AMSS, Chinese Academy of Sciences, Beijing 100190, People's Republic of China (dmcao@amt.ac.cn)
Wei Dai
Affiliation:
School of Mathematics and Systems Science, Beihang University (BUAA), Beijing 100191, People's Republic of China (weidai@buaa.edu.cn)
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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-existence of nontrivial nonnegative classical solutions in the subcritical cases 0 < γ < 1. As a consequence, we also derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities.

Keywords

bi-harmonicnonnegative solutionsLiouville type theoremsradial symmetryHartree type nonlinearitymethods of moving planes

MSC classification

Primary: 35J91: Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
Secondary: 35B06: Symmetries, invariants, etc. 35B53: Liouville theorems, Phragmén-Lindelöf theorems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 4 , August 2019 , pp. 979 - 994
Copyright
Copyright © Royal Society of Edinburgh 2018 

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