Differential and Integral Equations

Asymptotics of radial oscillatory solutions of semilinear elliptic equations

Xinfu Chen and Guozhen Lu

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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.

Article information

Differential Integral Equations, Volume 14, Number 11 (2001), 1367-1380.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations
Secondary: 34C10: Oscillation theory, zeros, disconjugacy and comparison theory 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35B40: Asymptotic behavior of solutions 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)


Chen, Xinfu; Lu, Guozhen. Asymptotics of radial oscillatory solutions of semilinear elliptic equations. Differential Integral Equations 14 (2001), no. 11, 1367--1380. https://projecteuclid.org/euclid.die/1356123029

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