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

TL;DR

This paper establishes a unilateral global bifurcation theorem for quasilinear elliptic problems with sign-changing weights and applies it to prove the existence of nodal and one-sign solutions, advancing understanding of nonlinear PDEs.

Contribution

It introduces a new unilateral bifurcation result for p-Laplacian problems with sign-changing weights and applies it to find nodal and one-sign solutions.

Findings

Existence of bifurcation points at eigenvalues

Presence of two unbounded solution continua

Existence results for nodal and one-sign solutions

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 Dr\'{a}bek 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.