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

Guowei Dai; Ruyun Ma; Haiyan Wang

doi:10.3934/cpaa.2013.12.2839

Article Contents

2013, Volume 12, Issue 6: 2839-2872. Doi: 10.3934/cpaa.2013.12.2839

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

1.
Department of Mathematics, Northwest Normal University, Lanzhou, 730070
2.
Division of Mathematical and Natural Sciences, Arizona State University, Phoenix, AZ 85069-7100

Received: November 30, 2012

Revised: January 31, 2013

Published: October 2013

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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 $\lambda_0^+$ and $\lambda_0^-$. Furthermore, under some natural hypotheses on perturbation function, we show that $(\lambda_0^\nu,0)$ is a bifurcation point of the above problems and there are two distinct unbounded sub-continua $C_\nu^{+}$ and $C_\nu^{-}$, consisting of the continuum $C_\nu$ emanating from $(\lambda_0^\nu, 0)$, where $\nu\in\{+,-\}$. 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 $\lambda$ are also studied.

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Mathematics Subject Classification: Primary: 34B18, 34C23; Secondary: 34D23, 34L05.

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