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

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 ${\mathcal{C}}_{\nu }^{+}$ and ${\mathcal{C}}_{\nu }^{-}$, consisting of the continuum ${\mathcal{C}}_{\nu }$ 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.

35 pages

References in corpus (1)

• The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations