Classification of entire solutions of $(-\Delta )^N u + u^{-(4N-1)}= 0$ with exact linear growth at infinity in $\mathbf {R}^{2N-1}$
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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 \[ u(x) = \int _{\mathbf {R}^{2N-1}} {|x - y|{u^{-(4N-1)}}(y)dy} \] up to a multiple constant and hence take the following form: \[ u(x) = (1+|x|^2)^{1/2} \] 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 $\mathbf {R}^{2N-1}$.References
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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
- 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
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 2585-2600
- MSC (2010): Primary 35B45, 35J40, 35J60
- DOI: https://doi.org/10.1090/proc/13960
- MathSciNet review: 3778160