On finding bifurcations for non-variational elliptic systems by the extended quotients method

TL;DR

This paper introduces a new method for detecting bifurcations in nonlinear elliptic systems using extended quotients, providing a variational approach to identify saddle-node bifurcation points and stability thresholds.

Contribution

The paper presents a novel extended quotients method for bifurcation detection in non-variational elliptic systems, including a variational formula for saddle-node bifurcations.

Findings

Identified the saddle-node bifurcation point for convex-concave elliptic equations.

Established a threshold value for the existence of stable positive solutions.

Validated the method through application to specific elliptic equations.

Abstract

We develop a novel method for finding bifurcations for nonlinear systems of equations based on directly finding bifurcations through saddle points of extended quotients. The method is applied to find the saddle-node bifurcation point for elliptic equations with the nonlinearity of the general convex-concave type. The main result justifies the variational formula for the detection of the maximum saddle-node type bifurcation point of stable positive solutions. As a consequence, a precise threshold value separating the interval of the existence of stable positive solutions is established.