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

Authors:
J. Karátson;P. L. Simon
Affiliations:
Department of Applied Analysis, Eötvös Loránd University, Budapest, Hungary;Department of Applied Analysis, Eötvös Loránd University, Budapest, Hungary
Venue:
Nonlinear Analysis: Theory, Methods & Applications
Year:
2003

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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 nondegenerate 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.