DOI: 10.14232/ejqtde.2024.1.67 ISSN: 1417-3875

Structures and evolution of bifurcation diagrams for a multiparameter p-Laplacian Dirichlet problem

Tsung-Yi Hsieh, Shin-Hwa Wang

We study the multiparameter p-Laplacian Dirichlet problem { ( φ p ( u ( x ) ) ) + λ ( k u p 1 + i = 1 m a i u q i ) μ j = 1 n b j u r j = 0 ,   1 < x < 1 , u ( 1 ) = u ( 1 ) = 0 , where p > 1 , φ p ( y ) = | y | p 2 y , ( φ p ( u ) ) is the one-dimensional p-Laplacian, λ > 0 and μ 0 are two bifurcation parameters. We assume that k 0 , , 0 < p 1 < q 1 < q 2 < < q m < r 1 < r 2 < < r n , for j = 1 , 2 , , n . We mainly prove that, on the ( λ , u ) -plane, the bifurcation diagram consists of a strictly decreasing curve for μ = 0 , and always consists of a -shaped curve for fixed μ > 0 . We then study the structures and evolution of the bifurcation diagrams with varying μ 0 .

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