Eigenvalues, bifurcation and one-sign solutions for the periodic p-Laplacian

2012/07/27 by Guowei Dai, Haiyan Wang, Dai, Guowei +1
Mathematics · Physics and Astronomy · #34B18 #34C23 #34D23 #34L05 #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1207.6670

openalex publication_date 2012/07/27 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

In this paper, we establish a unilateral global bifurcation result for a class of quasilinear periodic boundary problems with a sign-changing weight. By the Ljusternik-Schnirelmann theory, we first study the spectrum of the periodic p-Laplacian with the sign-changing weight. In particular, we show that there exist two simple, isolated, principal eigenvalues λ₀⁺ and λ₀^-. Furthermore, under some natural hypotheses on perturbation function, we show that (λ₀^ν,0) is a bifurcation point of the above problems and there are two distinct unbounded sub-continua \mathscrC_ν⁺ and \mathscrC_ν^-, consisting of the continuum \mathscrC_ν emanating from (λ₀^ν, 0), where ν∈\+,-\. As an application of the above result, we study the existence of one-sign solutions for a class of quasilinear periodic boundary problems with the sign-changing weight. Moreover, the uniqueness of one-sign solutions and the dependence of solutions on the parameter λ are also studied.

Citations

• The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations

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