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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Qualitative properties of entire radial solutions for a biharmonic equation with supercritical nonlinearity


Authors: Zongming Guo and Juncheng Wei
Journal: Proc. Amer. Math. Soc. 138 (2010), 3957-3964
MSC (2010): Primary 35J30; Secondary 35B08, 35B33
Published electronically: May 6, 2010
MathSciNet review: 2679617
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Abstract: We study some qualitative properties of entire positive radial solutions of the supercritical semilinear biharmonic equation:

$\displaystyle \Delta^2 u=u^p \;\;$   in $ \mathbb{R}^n$$\displaystyle , \;\; n \geq 5, \;\; p>\frac{n+4}{n-4}. \leqno(*)$

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
Email: wei@math.cuhk.edu.hk

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10374-8
PII: S 0002-9939(10)10374-8
Keywords: Entire radial solutions, biharmonic equations, singular solution
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
Article copyright: © Copyright 2010 American Mathematical Society