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2001 Asymptotics of radial oscillatory solutions of semilinear elliptic equations
Xinfu Chen, Guozhen Lu
Differential Integral Equations 14(11): 1367-1380 (2001).

Abstract

We study radially symmetric oscillatory solutions of semilinear elliptic equations of the form $$ \Delta u + \phi(|x|,u)=0\quad\hbox{in } \ R^n \ (n\geq 2)$$ where $\phi(r,u)$ is a nonnegative function having the form $\sum_i \! c_i r^{\nu_i} |u|^{p_i-1} u$ with $c_i>0$. Under certain resrictions on the exponents $\nu_i$ and $p_i$ (roughly speaking, $2\nu_i+n+2\geq (2-n)p_i$ for all $i$ where a strict inequality holds for at least one $i$), we show that all radial solutions must oscillate, i.e., change their signs infinitely many times. Moreover, we provide accurate estimates on the frequencies and amplitudes of these oscillatory solutions. These results are sharp in the sense that positive solutions exist when restrictions on these exponents are removed.

Citation

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Xinfu Chen. Guozhen Lu. "Asymptotics of radial oscillatory solutions of semilinear elliptic equations." Differential Integral Equations 14 (11) 1367 - 1380, 2001.

Information

Published: 2001
First available in Project Euclid: 21 December 2012

zbMATH: 1021.35036
MathSciNet: MR1859611

Subjects:
Primary: 35J60
Secondary: 34C10, 35B05, 35B40, 42B20

Rights: Copyright © 2001 Khayyam Publishing, Inc.

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Vol.14 • No. 11 • 2001
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