Bifurcation and Multiplicity Results for Elliptic Problems with Subcritical Nonlinearity on the Boundary

TL;DR

This paper investigates the bifurcation and multiplicity of positive solutions for elliptic boundary value problems with nonlinear boundary conditions involving subcritical growth, using degree theory and continuation methods.

Contribution

It introduces new bifurcation results for elliptic problems with nonlinear boundary conditions, including global bifurcation and solution multiplicity analysis.

Findings

Existence of a connected branch of positive solutions bifurcating from infinity as the parameter approaches zero.

Global bifurcation results under additional conditions near zero.

Discussion of the number of positive solutions relative to the bifurcation parameter.

Abstract

We consider an elliptic problem with nonlinear boundary condition involving nonlinearity with superlinear and subcritical growth at infinity and a bifurcation parameter as a factor. We use re-scaling method, degree theory and continuation theorem to prove that there exists a connected branch of positive solutions bifurcating from infinity when the parameter goes to zero. Moreover, if the nonlinearity satisfies additional conditions near zero, we establish a global bifurcation result, and discuss the number of positive solution(s) with respect to the parameter using bifurcation theory and degree theory.