On uniqueness of positive solutions of nonlinear differential equations. (English) Zbl 0904.34015
The authors consider existence and uniqueness of a positive solution to the problem
\[
u''+ g(x)u^\gamma(x)= 0,\quad 0<x<\infty
\]
with \(u(0)= 0\) and \(\lim_{x\to\infty} u(x)\) is a positive constant.
It is shown that for \(\gamma> 1\) there is at most one positive solution \(u\), under the condition that \({\gamma+ 3\over 2}+ {xg'\over g}\) has only a finite number of zeros and there is a positive solution with positive Pohozhaev function. If moreover \(g'(x)> 0\) everywhere then uniqueness follows.
It is shown that for \(\gamma> 1\) there is at most one positive solution \(u\), under the condition that \({\gamma+ 3\over 2}+ {xg'\over g}\) has only a finite number of zeros and there is a positive solution with positive Pohozhaev function. If moreover \(g'(x)> 0\) everywhere then uniqueness follows.
Reviewer: A.Barmistrova (Chelyabinsk)
MSC:
34B15 | Nonlinear boundary value problems for ordinary differential equations |