Mathematics

Estimates of positive nontrivial solutions of a differential equation with power nonlinearity

D. A. Bezukhov

Lomonosov Moscow State University, 1, Leninskie Gory, Moscow, 119991, Russian Federation

Abstract: Differential equations
$$ y^{[n]}=r_n(x)\frac{d}{dx}\left( r_{n-1}(x)\frac{d}{dx}\left(\ldots\left( r_0(x) y\frac{}{} \right)\right)\ldots\right)=(-1)^np(x)|y|^k $$
and
$$ y^{(n)}=(-1)^np(x)|y|^k $$
with power nonlinearity are considered. Solutions which are defined in some neighborhood of plus infinity are called proper solutions. It is proved that proper solution to the equation is kneser solution, which means that solution and it’s quasiderivatives change their signs and tend to zero. The integral representation for proper solutions is proved. Upper estimates for solution and it’s quasiderivatives for proper solutions with maximal interval of existence is positive semiaxis to the equation with quasiderivative are proved. Upper and lower estimates of solution and it’s derivatives for proper solutions with maximal interval of existence is positive semiaxis to the equation with derivative are proved.

Keywords: Emden–Fowler equation, estimates of solutions to the nonlinear defferential equation, quasiderivative.

UDC: 517.925.54

Received: 08.07.2015

Bibliographic databases:

•