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

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 $λ_0^+$ and $λ_0^-$. Furthermore, under some natural hypotheses on perturbation function, we show that $(λ_0^ν,0)$ is a bifurcation point of the above problems and there are two distinct unbounded sub-continua $\mathscr{C}_ν^{+}$ and $\mathscr{C}_ν^{-}$, consisting of the continuum $\mathscr{C}_ν$ emanating from $(λ_0^ν, 0)$, where $ν\in\{+,-\}$. 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 $λ$ are also studied.