Sign-changing solutions to singular second-order boundary value problems. (Q5954480)

The authors' present an general framework for sign-changing solutions of NEWLINE\[NEWLINEu''+ p(x)|u|^{-\gamma}= 0\quad\text{on }(a,b)\tag{1}NEWLINE\]NEWLINE and its multidimensional analogue NEWLINE\[NEWLINE\begin{gathered} \Delta u+ p(x) u^{(-\gamma)}= 0\quad\text{in }\Omega,\\ u= 0\quad\text{on }\partial\Omega\end{gathered}\tag{2}NEWLINE\]NEWLINE on a bounded, smooth domain \(\Omega\subset\mathbb{R}^N\). Here \(s^{(-\gamma)}\) stands for the odd power-function \(|s|^{-\gamma}\) signs. To this end the authors use erlatiely prcise determination of the boundary behaviour of positive solution of (2). Sign-changing principal-value solutions are constructed by gluing together solutions of one sign.