Positive symmetric solutions of singular semipositone boundary value problems

M. Rudd, University of Idaho, Moscow, ID, U.S.A.
C. C. Tisdell, The University of New South Wales, Sidney, Australia

E. J. Qualitative Theory of Diff. Equ., Spec. Ed. I, 2009 No. 24., pp. 1-10.

Abstract: Using the method of upper and lower solutions, we prove that the singular boundary value problem,
\[
-u'' = f(u) ~ u^{-\alpha} \quad \textrm{in} \quad (0, 1), \quad u'(0) = 0 = u(1) \, ,
\]
has a positive solution when $0 < \alpha < 1$ and $f : \R \to \R$ is an appropriate nonlinearity that is bounded below; in particular, we allow $f$ to satisfy the semipositone condition $f(0) < 0$. The main difficulty of this approach is obtaining a positive subsolution, which we accomplish by piecing together solutions of two auxiliary problems. Interestingly, one of these auxiliary problems relies on a novel fixed-point formulation that allows a direct application of Schauder's fixed-point theorem.

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