Homoclinic Orbits And Boundary Value Problem For A Class Of Fourth-order Differential Equations

This dissertation mainly concerns with the existence, multiplicity, nonexistenceof solutions of the boundary value problem and the existence of homoclinic orbitsfor a class of fourth order super-linear differential equations. By making use of theclassical variational techniques and critical point theory, the existence and multiplicityof periodic solutions of a single equation in literature are extended to that of equations,and also the cubic growth of nonlinear term is extended to a general form of super-linear growth. It is composed of three chapters.Chapter 1 concentrates on the brief introduction of historic background, signif-icance for the problems under consideration, preliminaries and main results of thisdissertation.In chapter 2, firstly, by establishing the corresponding variational structure, theproblem of finding solutions of the differential equations is reduced to that of seekingcritical points of the corresponding functional. And then by using Clark theorem,multiplicity of the equations is studied, Finally the nonexistence of the equations isalso investigated and by using the symmetric Mountain-Pass theorem, the existenceof infinitely many pairs of solutions are obtained.In Chapter 3 Mountain-Pass theorem is emploited to establish the existence ofhomoclinic orbits for a class of fourth-order differential equations under the periodicconditions.