Unilateral global bifurcation and nodal solutions for the p-Laplacian with sign-changing weight

Abstract

In this paper, we shall establish a Dancer-type unilateral global bifurcation result for a class of quasilinear elliptic problems with sign-changing weight. Under some natural hypotheses on perturbation function, we show that is a bifurcation point of the above problems and there are two distinct unbounded continua, and , consisting of the bifurcation branch from , where is the th positive or negative eigenvalue of the linear problem corresponding to the above problems, . As applications of the above unilateral global bifurcation result, we study the existence of nodal solutions for a class of quasilinear elliptic problems with sign-changing weight. Moreover, based on the bifurcation result of Drábek and Huang (1997) Dai G, Ma R. Unilateral global bifurcation phenomena and nodal solutions for -Laplacian. J. Differ. Equ. 2012;252:2448–2468., we study the existence of one-sign solutions for a class of high-dimensional quasilinear elliptic problems with sign-changing weight.