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.