Multiplicity for a strongly singular quasilinear problem via bifurcation theory

TL;DR

This paper investigates a complex p-Laplacian elliptic problem with singular and superlinear nonlinearities, using bifurcation theory to establish the existence of an unbounded branch of positive solutions and analyze their multiplicity.

Contribution

It introduces a novel application of bifurcation theory combined with approximation and sub-supersolution methods to study strongly singular quasilinear problems.

Findings

Existence of an unbounded branch of positive solutions

Bifurcation from infinity at zero parameter value

Intervals of existence, nonexistence, and multiplicity of solutions

Abstract

A $p$-Laplacian elliptic problem in the presence of both strongly singular and $(p-1)$-superlinear nonlinearities is considered. We employ bifurcation theory, approximation techniques and sub-supersolution method to establish the existence of an unbounded branch of positive solutions, which is bounded in positive $\lambda-$direction and bifurcates from infinity at $\lambda=0$. As consequence of the bifurcation result, we determine intervals of existence, nonexistence and, in particular cases, global multiplicity.