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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Qualitative properties of entire radial solutions for a biharmonic equation with supercritical nonlinearity
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by Zongming Guo and Juncheng Wei PDF
Proc. Amer. Math. Soc. 138 (2010), 3957-3964 Request permission

Abstract:

We study some qualitative properties of entire positive radial solutions of the supercritical semilinear biharmonic equation: \[ \Delta ^2 u=u^p \;\; \text {in $\mathbb {R}^n$}, \;\; n \geq 5, \;\; p>\frac {n+4}{n-4}. \tag {*} \] It is known from a paper by Gazzola and Grunau that there is a critical value $p_c>(n+4)/(n-4)$ of $(*)$ for $n \geq 13$ and that $(*)$ has a singular solution $u_s (r)= K_0^{1/(p-1)} r^{-4/(p-1)}$. We show that for $5 \leq n \leq 12$ or $n \geq 13$ and $p<p_c$, any regular positive radial entire solution $u$ of $(*)$ intersects with $u_s (r)$ infinitely many times. On the other hand, if $n \geq 13$ and $p \geq p_c$, then $u(r)<u_s (r)$ for all $r>0$. Moreover, the solutions are strictly ordered with respect to the initial value $a=u(0)$.
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Additional Information
  • Zongming Guo
  • Affiliation: Department of Mathematics, Henan Normal University, Xinxiang, 453007, People’s Republic of China
  • Email: guozm@public.xxptt.ha.cn
  • Juncheng Wei
  • Affiliation: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong
  • MR Author ID: 339847
  • ORCID: 0000-0001-5262-477X
  • Email: wei@math.cuhk.edu.hk
  • Received by editor(s): August 5, 2009
  • Received by editor(s) in revised form: January 12, 2010
  • Published electronically: May 6, 2010
  • Communicated by: Yingfei Yi
  • © Copyright 2010 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 138 (2010), 3957-3964
  • MSC (2010): Primary 35J30; Secondary 35B08, 35B33
  • DOI: https://doi.org/10.1090/S0002-9939-10-10374-8
  • MathSciNet review: 2679617