Advances in Differential Equations

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

P. J. McKenna and W. Reichel

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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.

Article information

Adv. Differential Equations, Volume 6, Number 4 (2001), 441-460.

First available in Project Euclid: 2 January 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations
Secondary: 34B15: Nonlinear boundary value problems 35B15: Almost and pseudo-almost periodic solutions 35B40: Asymptotic behavior of solutions


McKenna, P. J.; Reichel, W. Sign-changing solutions to singular second-order boundary value problems. Adv. Differential Equations 6 (2001), no. 4, 441--460.

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