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Classification of entire solutions of $ (-\Delta)^N u + u^{-(4N-1)}= 0$ with exact linear growth at infinity in $ \mathbf{R}^{2N-1}$


Author: Quốc Anh Ngô
Journal: Proc. Amer. Math. Soc. 146 (2018), 2585-2600
MSC (2010): Primary 35B45, 35J40, 35J60
DOI: https://doi.org/10.1090/proc/13960
Published electronically: February 28, 2018
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Abstract: In this paper, we study global positive $ C^{2N}$-solutions of the geometrically interesting equation $ (-\Delta )^N u + u^{-(4N-1)}= 0$ in $ \mathbf {R}^{2N-1}$. Using the sub poly-harmonic property for positive $ C^{2N}$-solutions of the differential inequality $ (-\Delta )^N u < 0$ in $ \mathbf {R}^{2N-1}$, we prove that any $ C^{2N}$-solution $ u$ of the equation having linear growth at infinity must satisfy the integral equation

$\displaystyle u(x) = \int _{\mathbf {R}^{2N-1}} {\vert x - y\vert{u^{-(4N-1)}}(y)dy} $

up to a multiple constant and hence take the following form:

$\displaystyle u(x) = (1+\vert x\vert^2)^{1/2} $

in $ \mathbf {R}^{2N-1}$ up to dilations and translations. We also provide several non-existence results for positive $ C^{2N}$-solutions of $ (-\Delta )^N u = u^{-(4N-1)}$ in $ \mathbf {R}^{2N-1}$.

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Additional Information

Quốc Anh Ngô
Affiliation: Institute of Research and Development, Duy Tân University, Dà Nǎng, Viêt Nam —and— Department of Mathematics, College of Science, Viêt Nam National University, Hà Nôi, Viêt Nam
Email: nqanh@vnu.edu.vn, bookworm_vn@yahoo.com

DOI: https://doi.org/10.1090/proc/13960
Keywords: High-order elliptic equation, negative exponent, $Q$-curvature, radially symmetrical, linear growth at infinity
Received by editor(s): July 13, 2017
Received by editor(s) in revised form: September 6, 2017
Published electronically: February 28, 2018
Communicated by: Guofang Wei
Article copyright: © Copyright 2018 American Mathematical Society

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