Bifurcation and one-sign solutions of the $p$-Laplacian involving a nonlinearity with zeros

TL;DR

This paper investigates the existence and multiplicity of positive or negative solutions for the p-Laplacian with nonlinearities that have zeros, using bifurcation theory and topological methods.

Contribution

It introduces a bifurcation theorem from infinity for nonlinear operators and applies it to analyze solutions of the p-Laplacian with nonlinearities involving zeros.

Findings

Established a bifurcation theorem from infinity for nonlinear operators

Proved existence and multiplicity of one-sign solutions for the p-Laplacian

Developed topological results involving superior limits for superlinear cases

Abstract

In this paper, we use bifurcation method to investigate the existence and multiplicity of one-sign solutions of the $p$-Laplacian involving a linear/superlinear nonlinearity with zeros. To do this, we first establish a bifurcation theorem from infinity for nonlinear operator equation with homogeneous operator. To deal with the superlinear case, we establish several topological results involving superior limit.