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 lambda(+)(0) and lambda(-)(0). Furthermore, under some natural hypotheses on perturbation function, we show that (lambda(v)(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 C-v, emanating from (lambda(v)(0), 0), where v epsilon {+,-}. 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 A are also studied.