Fixed points of maps and nontrivial weak solutions to a class of nonlinear strongly coupled elliptic systems

TL;DR

This paper investigates fixed points of nonlinear maps and applies the results to prove the existence of nontrivial weak solutions for certain strongly coupled elliptic systems, highlighting a bifurcation phenomenon caused by cross-diffusion.

Contribution

It introduces new methods for analyzing fixed points of nonlinear maps and demonstrates their application to establish solutions for complex elliptic systems.

Findings

Existence of nontrivial weak solutions for specific elliptic systems

Identification of bifurcation phenomena due to cross-diffusion

Analysis of local indices at fixed points

Abstract

Local indices at isolated fixed points of a differentiable compact nonlinear map $T$ on Banach spaces will be discussed. These results are applied to establish the existence of nontrivial solutions. As an example, the existence of nontrivial weak solutions to a class of nonlinear strongly coupled elliptic systems of two equations will also be studied. A bifurcation phenomenon due to cross-diffusion is reported.