Sign-changing solutions to singular second-order boundary value problems

Abstract

We consider elliptic boundary value problems of the form $u''+p(x)|u|^{-\gamma}{{\rm sign}\:} u=0$ and $\Delta u + p(x)|u|^{-\gamma}{{\rm sign}\:} u=0$ for $\gamma>1$ and $p>0$ on intervals in ${\mathbb {R}}$ and bounded domains in ${\mathbb {R}}^n$. Prescribed vanishing Dirichlet boundary data and the concept of sign-changing solutions make the problem singular both on the boundary and in the interior of the underlying domain. We introduce a solution concept for sign-changing solutions based on the distributional principal value, and we show existence of such solutions. A principal part of our analysis is a (relatively) accurate description of the boundary behavior of positive solutions. Sign-changing principal-value solutions are constructed by gluing together solutions of one sign.