Singular eigenvalue problem for the Emden-Fowler equation. Existence of a solution with given number of zeroes (Q2755263)

The authors consider the singular boundary value problem NEWLINE\[NEWLINE{d^2y\over dx^2} +{a\over x}{dy\over dx}+{|y|^{k}y\over x}=\lambda^2 y, \quad y(0)=1, \;y(+\infty)=0,NEWLINE\]NEWLINE and reduce it to the problem NEWLINE\[NEWLINEy''+x^{-1}(ay'+|y|^{k} y)=y, \quad y(0)=c,\;y(+\infty)=0.NEWLINE\]NEWLINE Let us denote by \(y(x,c)\) a solution to the second problem. The main result of the article is the following:NEWLINENEWLINENEWLINELet \(\max(0,1-1/k)\leq a<1+2/k\). Then for arbitrary \(l\geq 1, l\in \mathbb{N}\), there exists such \(\sigma_{l}>0\) that the function \(y(x,\sigma_{l})\) has \(l\) zeroes on \((0,+\infty)\) and satisfies the second problem with \(c=\sigma_{l}\).