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

TL;DR

This paper investigates the spectral properties and bifurcation phenomena of the periodic $p$-Laplacian with sign-changing weights, establishing the existence and uniqueness of one-sign solutions and their dependence on parameters.

Contribution

It introduces a new unilateral global bifurcation result for quasilinear periodic problems with sign-changing weights, identifying principal eigenvalues and solution continua.

Findings

Existence of two principal eigenvalues $a0a0a0a0a0a0a0a0a0a0a0$ with sign-changing weights.

Existence of two unbounded solution continua bifurcating from eigenvalues.

Results on the uniqueness and parameter dependence of one-sign solutions.

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