A COMPLETE CLASSIFICATION OF BIFURCATION DIAGRAMS OF AP-LAPLACIAN DIRICHLET PROBLEM II. GENERALIZED NONLINEARITIES

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Titel: A COMPLETE CLASSIFICATION OF BIFURCATION DIAGRAMS OF AP-LAPLACIAN DIRICHLET PROBLEM II. GENERALIZED NONLINEARITIES

Beteiligte: Wang, Feng-Lin; Wang, Shin-Hwa

Erschienen: Mathematical Society of the Republic of China, 2012

Sprache: Englisch

ISSN: 1027-5487; 2224-6851

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Beschreibung: Abstract We study the bifurcation diagrams of classical positive solutionsuwith ‖u‖∞∈ (0, ∞) of thep-Laplacian Dirichlet problem { ( φ p ( u ′ ( x ) ) ) ′ + λ f q , r ( u ( x ) ) = 0 , − 1 < x < 1 , u ( − 1 ) = 0 = u ( 1 ) , wherep> 1,φp (y) = |y| p−2 y, (φp (u′))′ is the one-dimensionalp-Laplacian, λ > 0 is a bifurcation parameter, and f q , r ( u ) = { | 1 − u | q , if 0 < u ≤ 1 , | 1 − u | r , if u > 1 , with positive constantsqandr. We give explicit formulas of bifurcation curves of classical positive solutions on the (λ, ‖u‖∞)-plane. More importantly, for different (p, q, r), we give a complete classification of all bifurcation diagrams. Hence we are able to determine the (exact) multiplicity of classical positive solutions for each (p, q, r, λ). Our results generalize the results of Leeet al.[J. Math. Anal. Appl.,330(2007), 276-290] with nonlinearityfq,r generalized fromq=r> 0 toq, r> 0. 2010Mathematics Subject Classification: 34B18, 35B32. Key words and phrases: Bifurcation diagram, Positive solution, Exact multiplicity,p-Laplacian, Time map.

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