The authors consider the nonlinear boundary-value problem on the half- line (1) \(y''+ {N- 1\over x} y'= f(y)\), \(x> 0\), \(y'(0)= y(+\infty)= 0\), where \(f\) is a smooth function, \(f'(0)> 0\), \(N> 1\), and establish a theorem on the existence of solutions with any number of roots for the case when \(f(y)\) is an odd function which satisfies the condition \(\lim_{y\to \infty} {f(y)\over y}= \lambda\leq 0\) and has exactly one positive root. A great bibliography of papers where problem (1) is considered is given in this paper.