Morse theory for a fourth order elliptic equation with exponential nonlinearity

TL;DR

This paper develops a new degree counting formula for a fourth order elliptic PDE with exponential nonlinearity, using topological methods, complementing previous blow-up analysis approaches.

Contribution

It introduces a Poincare'-Hopf type theorem and a geometric degree counting formula for the problem, providing an alternative to blow-up estimates.

Findings

Established a Leray Schauder degree for the problem

Proved a Poincare'-Hopf type theorem for the PDE

Derived a geometric degree counting formula

Abstract

In this paper we compute the Leray Schauder degree for a fourth order elliptic boundary value problem with exponential nonlinearity and Navier boundary condition. This will be made by proving a Poincare'-Hopf type theorem. Moreover by using this result, together with some quantitative results about the formal set of barycenters, we are able to establish a direct and geometrically clear degree counting formula for our problem. We remark that this formula has been proven with complete different methods by Lin Wei and Wang by using blow-up type estimates.