## Differential and Integral Equations

### Asymptotics of radial oscillatory solutions of semilinear elliptic equations

#### 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.

#### Article information

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

Dates
First available in Project Euclid: 21 December 2012