Morse Theory And Its Application To Semi-Linear Elliptic Equations

Calculus of variations and partial differential equations(PDEs)are the significant research fields in modern mathematics.Theses research fields are not only applied to some branches of mathematics,such as differential geom-etry and harmonic analysis,but also are evident in the general applications to physics,mechanics and biology.Calculus of variations mainly research the existence of maximum and minimum of functional,which means that finding critical points of some functionals owning corresponding critical values.These critical points can be some coordinates,paths and curves or surfaces.There-for,these topics are also called critical point theory,and Morse theory is the important content of calculus of variations,meanwhile,this theory can provide corresponding methods to find some critical points.M.Morse is the first per-son who works in global variational analysis and proposes Morse theory,and then J.Milnor summarize and improve Morse theory.Kung-ching Chang de-votes himself in developing infinite dimensional Morse theory and applying this theory to PDEs.Wang Zhiqiang proposes equivariant Morse theory based on infinite dimensional Morse theory.In this thesis,we research the isolated crit-ical points of some functions defined on the manifolds under general boundary conditions with compact Lie groups action and work out equivariant Morse theory on these manifolds.According to this work,we study the existence and multiplicity of group-equivariant solutions to the Dirichlet problems for semi-linear elliptic equation.This thesis is mainly divided four chapters to illustrate what we study.In chapter 1,we introduce development background of calculus of variations,the frame of our work and significance of our study.Then in chapter 2,we briefly illustrate some relevant concepts of infinite dimensional Morse theory and some theorems in calculus of variations and PDEs.These concepts and theorems can be applied to the last two chapters.Chapter 3 and chapter 4 are the most essential contents of the thesis.In chapter 3,we mainly talk about the equivariant Morse theory based on the manifolds with group-invariant(G-invariant)general boundary conditions.We divide this chapter into four sections to discuss.In the first section,we illus-trate some corresponding concepts on group action.From these concepts,we propose our research objects including G-orbit,orbit space,G-invariant space and G-invariant functions,as well as group-equivariant map(G-equivariant map).In the second section,we mainly study about the general boundary conditions with group action and reasonability of outward normal unit on the boundary of the manifold we study.In the third section,we construct the G-cohomology of the manifolds with general boundary conditions under group action,including G-cohomology,G-critical groups,Morse index,and Morse type numbers.In addition,this cohomology is based on fibration with group action,so we shall talk about the fibration with group action,then propose G-cohomology.At last,we prove the most two important theorems,Equivariant Inequalities and Equivariant Morse Handle Body Theorem,in detail.Chapter 4 provide some topics about the existence of equivariant solutions and the multiplicity of solutions to the Dirichlet problems of semi-linear elliptic equations.We also divide this chapter into four parts.In the first part,we propose the Sobolev space consisting of equivariant maps and its properties including Poincare equalities and Sobolev inequalities;In the second part,we study the invariance of elliptic equations under compact Lie group action.We conclude that the reasonability of the existence of equivariant solutions to Laplace equations by the invariance of Laplace equations under compact Lie group with special orthogonal representation.Then we discuss the multiplicity of elliptic solutions to the second order semi-linear elliptic equations in the third part.We choose two classical sorts of elliptic PDEs to study.One is super-linear elliptic equations,we prove that there exist at least three distinct equivariant solutions to Dirichlet problems for super-linear elliptic equations by equivariant Morse equalities stated in Chapter 3.The other is elliptic Euler-Lagrange equations with homogeneous Dirichlet boundary conditions.We conclude that there exist infinitely many solutions to these elliptic equations via mountain pass lemma and the existence of critical points for even functional theorem.At last section,we state some examples to verify our conclusions in the third part of this chapter.