Exact multiplicity for degenerate two-point boundary value problems with p-convex nonlinearity

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Abstract

The exact number of positive solutions of a degenerate quasilinear two-point boundary value problem is investigated. For the generalization of earlier results concerning the non-degenerate case with convex nonlinearity, suitably defined p-convex nonlinearities are considered. Strictly p-convex C2 functions having a non-negative root are classified according to the shape of the bifurcation diagram of positive solutions versus the length of the interval. We have uniqueness when f(0)≤0 and 0, 1 or 2 solutions when f(0)>0 (similarly to the non-degenerate case), provided that the number of solutions is finite. However, now there may also occur a continuum of solutions, connected to a dead core type phenomenon. The proof of our results relies on the shooting method for the characterization of the shape of the time-map. In contrast to the non-degenerate case, the shooting method does not determine directly the number of solutions, owing to the lack of uniqueness of the corresponding IVP. Exact conditions on the uniqueness of the IVP and, in the case of non-uniqueness, the number and types of its local solutions are given. Based on this, all the positive solutions of the BVP can be compiled.

Original languageEnglish
Pages (from-to)1569-1590
Number of pages22
JournalNonlinear Analysis, Theory, Methods and Applications
Volume52
Issue number6
DOIs
Publication statusPublished - Jun 1 2003

Keywords

  • Bifurcation diagram of positive solutions
  • Dead core
  • Degenerate two-point boundary value problem
  • Non-uniqueness of IVP
  • Time-map

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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