Existence of multiple solutions for a quasilinear elliptic problem

TL;DR

This paper proves the existence of multiple solutions for a quasilinear elliptic boundary value problem using bifurcation techniques, under conditions related to the p-derivative of the nonlinearity at zero and infinity.

Contribution

It introduces new bifurcation methods from zero and infinity to establish multiple solutions for the p-Laplace problem under specific derivative conditions.

Findings

Multiple solutions exist for the problem.

Unbounded branches of solutions are characterized.

Qualitative properties of solutions are provided.

Abstract

In this paper we prove the existence of multiple solutions for a quasilinear elliptic boundary value problem, when the p-derivative at zero and the p-derivative at infinity of the nonlinearity are greater than the first eigenvalue of the p-Laplace operator. Our proof uses bifurcation from infinity and bifurcation from zero to prove the existence of unbounded branches of positive solutions (resp. of negative solutions). We show the existence of multiple solutions and we provide qualitative properties of these solutions.