On the existence of nonoscillatory solutions to sublinear Emden-Fowler equations (Q2782926)

Here, the sublinear Emden-Fowler equation NEWLINE\[NEWLINE y''+a(x)|y|^{\gamma-1}y=0, \quad x>0,\tag{1}NEWLINE\]NEWLINE with \(0<\gamma<1\) and \(a\in C(\mathbb{R}^+,\mathbb{R}^+)\), is studied. The author presents sufficient conditions under which equation (1) has nonoscillatory solutions with arbitrary large zeros. This work completes a similar result from \textit{Z. Nehari} [Duke Math. J. 42, 183-189 (1975; Zbl 0385.34011)], where \(\gamma>1\) is considered. The main tools are a transformation into the equation NEWLINE\[NEWLINE\ddot{u}+\Bigl(g(t)|u|^{\gamma-1}-\frac{1}{4}\Bigr)u=0, \quad t>1,NEWLINE\]NEWLINE and the energy function \(G({u}(t))=\frac{\dot{u}^2(t)}{2}+ \frac{u^2(t)}{2}(\frac{2g(t)|u(t)|^{\gamma-1}}{\gamma+1} -\frac{1}{4})\).