## Arkiv för Matematik

- Ark. Mat.
- Volume 37, Number 2 (1999), 305-322.

### On Kneser solutions of higher order nonlinear ordinary differential equations

#### Abstract

The equation *x*^{(n)}(t)=(−1)^{n}│*x(t)*│^{k} with *k*>1 is considered. In the case *n*≦4 it is proved that solutions defined in a neighbourhood of infinity coincide with *C*(t−t_{0})^{−n/(k−1)}, where *C* is a constant depending only on *n* and *k*. In the general case such solutions are Kneser solutions and can be estimated from above and below by a constant times (*t−t*_{0})^{−n/(k−1)}. It is shown that they do not necessarily coincide with *C*(t−t_{0})^{−n/(k−1)}. This gives a negative answer to two conjectures posed by Kiguradze that Kneser solutions are determined by their value in a point and that blow-up solutions have prescribed asymptotics.

#### Dedication

Dedicated to Professor Vladimir Maz'ya on the occasion of his 60th birthday.

#### Note

The author was supported by the Swedish Natural Science Research Council (NFR) grant M-AA/MA 10879-304.

#### Article information

**Source**

Ark. Mat., Volume 37, Number 2 (1999), 305-322.

**Dates**

Received: 9 October 1997

Revised: 20 October 1998

First available in Project Euclid: 31 January 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.afm/1485898637

**Digital Object Identifier**

doi:10.1007/BF02412217

**Mathematical Reviews number (MathSciNet)**

MR1714766

**Zentralblatt MATH identifier**

1118.34317

**Rights**

1999 © Institut Mittag-Leffler

#### Citation

Kozlov, Vladimir A. On Kneser solutions of higher order nonlinear ordinary differential equations. Ark. Mat. 37 (1999), no. 2, 305--322. doi:10.1007/BF02412217. https://projecteuclid.org/euclid.afm/1485898637