Oscillations in a semi-linear singular Sturm–Liouville equation

Abstract

Abstract

We study the semi-linear Sturm–Liouville equation $$\left\{\begin{array}{c}-{\mathrm{\left(}{\mathit{x}}^{2\mathit{\alpha }}{\mathit{u}}^{\prime }\mathrm{\right)}}^{\prime }=\mathit{\lambda }\mathit{u}+|\mathit{u}{|}^{\mathit{p}-1}\mathit{u}\text{in }\mathrm{(}0\mathrm{,}1\mathrm{)}\mathrm{,}\hfill \\ \mathit{u}\mathrm{(}1\mathrm{)}=0\mathrm{,}\hfill \end{array}\right.$$ where $\mathit{\alpha }⩾1$ , $\mathit{p}\mathrm{>}1$ and λ are real parameters. We prove that all non-trivial solutions are oscillatory and unbounded as x approaches 0. Moreover, there exist $\mathit{\gamma }\mathrm{>}0$ and $\mathit{\delta }\mathrm{>}0$ such that any solution $\mathit{u}\mathrm{(}\mathit{x}\mathrm{)}$ resembles ${\mathit{x}}^{-\mathit{\gamma }}\mathit{\eta }\mathrm{(}{\mathit{x}}^{-\mathit{\delta }}\mathrm{)}$ near the origin, where η is a non-trivial periodic solution to the Emden–Fowler equation ${\mathit{\delta }}^2{\mathit{\eta }}^{″}+|\mathit{\eta }{|}^{\mathit{p}-1}\mathit{\eta }=0$ in $[0\mathrm{,}\infty \mathrm{)}$ .