Research Of Multiple Solutions For Several Asymptotically Linear Differential Equations

This thesis consists of five chapters.In Chapter1, we give a brief introduction to the background and preliminary knowledge.In Chapter2, we develop index theory for linear elliptic equations by Morse index. Thanks to this index theory, we investigate the existence and multiplicity of solutions to asymptotically linear elliptic equations with resonance.In Chapter3, we develop index theory for linear first order Hamiltonian systems by μ-index and the relative Morse index. Using dual variational method and convex analysis, we investigate the existence and multiplicity of solutions to asymptotically linear first order Hamiltonian systems.In Chapter4, we investigate an unbounded self-adjoint equation which possess only discrete spectrum. By using the index theory for the associated linear unbounded self-adjoint operation, we consider the existence and multiplicity for asymptotically linear self-adjoint equation. As applications, we investigate asymptotically linear Hamiltonian systems satisfying various boundary value conditions.In Chapter5, by using Morse index, we establish an index theory for a class of linear schrodinger equation, whose spectrum has finite number of eigenvalues below the infimum of the essential spectrum. By using this index theory, we consider the existence and multiplicity of solutions for asymptotically linear schrodinger equation.