The generalized Conley index and multiple solutions of semilinear elliptic problems

Yihong Du,E N Dancer

doi:10.1155/s108533759600005x

Abstract

We establish some framework so that the generalized Conley index can be easily used to study the multiple solution problem of semilinear elliptic boundary value problems. Both the parabolic flow and the gradient flow are used. Some examples are given to compare our approach here with other well-known methods. Our abstract results with parabolic flows may have applications to parabolic problems as well.

Highlights

We continue our efforts to show how the generalized Conley index developed by Rybakowski can be applied to multiple solution problems of semilinear elliptic equations
Our main purpose here is to set up some framework so that the generalized Conley index can be used to superlinear problems
We study in some detail a more general version of a semilinear elliptic problem with a combined concave and convex nonlinearity, which was studied in [2] and [1] recently

Summary

Introduction

We continue our efforts to show how the generalized Conley index developed by Rybakowski can be applied to multiple solution problems of semilinear elliptic equations. We need to establish proper compactness settings in the generalized Conley index for superlinear nonlinearities This is done by making use of the energy functionals, a priori estimates (for the parabolic flow) and P.S. condition (for the gradient flow). As a simple example to show that the generalized Conley index approach may give better results than the other well established methods, we apply our general results on the parabolic and gradient flows to a special case of (1.1), i.e., the following Neumann problem:. Since the generalized Conley index possesses continuation stability, it may be useful to consider the flow π for a single limiting equation as a limit of the flow π generated by the original system and use results on π to study π In this case, the flow η is difficult to use as there is in general no gradient flow for systems due to the fact that most natural reaction-diffusion systems lack variational structures.

The Parabolic Flow

The Gradient Flow

A Dirichlet Problem