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 γ+ 0 and γ- 0 . Further-more, under some natural hypotheses on perturbation function, we show that (γv 0 ; 0) is a bifurcation point of the above problems and there are two distinct unbounded sub-continua C+ v and C- v , consisting of the continuum Cv emanating from (γv 0 ; 0) , where v ∈ {+; -}. 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 γ are also studied.