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A complete classification of bifurcation diagrams of a p-Laplacian Dirichlet problem

✍ Scribed by Shin-Hwa Wang; Tzung-Shin Yeh

Publisher
Elsevier Science
Year
2006
Tongue
English
Weight
294 KB
Volume
64
Category
Article
ISSN
0362-546X

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✦ Synopsis

We study the bifurcation diagrams of positive solutions of the p-Laplacian Dirichlet problem

is the onedimensional p-Laplacian, and > 0 is a bifurcation parameter. We assume that functions g and h satisfy hypotheses (H1)-(H3). Under hypotheses (H1)-(H3), we give a complete classification of bifurcation diagrams, and we prove that, on the ( , u ∞ )-plane, each bifurcation diagram consists of exactly one curve which is either a monotone curve or has exactly one turning point where the curve turns to the right. Hence the problem has at most two positive solutions for each > 0. More precisely, we prove the exact multiplicity of positive solutions. In addition, for p = 2, we give interesting examples which show the evolution phenomena of bifurcation diagrams.

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