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 $({\mu }_k^{\nu }(p),0)$ is a bifurcation point of the above problems and there are two distinct unbounded continua, $({\mathcal{C}}_k^{\nu }{)}^{+}$ and $({\mathcal{C}}_k^{\nu }{)}^{-}$, consisting of the bifurcation branch ${\mathcal{C}}_k^{\nu }$ from $({\mu }_k^{\nu }(p),0)$, where ${\mu }_k^{\nu }(p)$ is the $k$-th positive or negative eigenvalue of the linear problem corresponding to the above problems, $\nu \in \{+,-\}$. As the 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) [\ref{DH}], we study the existence of one-sign solutions for a class of high dimensional quasilinear elliptic problems with sign-changing weight.