Homoclinic Solutions Of Hamiltonian Systems With P-laplace Operator

In this thesis, we consider the existence of homoclinic solutions for p-Laplacian Hamiltonian system via critical point theory, nil-boundary-value problems and variational methods, obtain several new conclusions and extend the correspond-ing results in the existed papers.The thesis is divided into four sections according to the contents.Chapter 1 briefly introduces the researched background of problems and the main work of the paper.Chapter 2 studies the second-order Hamiltonian system u(t)+▽W(t,u(t))＝0, and give new conditions for the existence of even homoclinic solutions, where t∈R,u∈Rn, and W∈C1(R×Rn, R) is T-periodic with respect to t, T>0.A homoclinic orbit is obtained as a limit of a certain sequence of nil-boundary-value problems which are obtained by the Mountain Pass theorem.Chapter 3 studies the p-Laplacian Hamiltonian system and obtain a new existence result of homoclinic solution, where t∈R,p>1,u∈Rn, f:R�'Rn is a continuous and bounded function, F∈C1(R×Rn,R) is Tperiodic with respect to t for T>0 and L:R�'Rn is a matrix valued function. The homoclinic solution is obtained as a limit of 2kT-periodic orbits of a sequence of second-order differential equations.In Chapter 4, the existence of homoclinic solutions for the p-Laplacian Hamil-tonian system is investigated under the local condition F(t,x)≥F(t,0)+b|x|u-a(t)|x|v for all (t, x)∈R×Rn, where p＞1,b＞0,μ＞v＞1 are constants and a:R�'R+ is a positive continuous function such that a∈L v/(μ-v)(R, R). The obtained result extends and improves the corresponding results in the existed papers. The homoclinic solution is given as a limit of 2kT-periodic orbits of a sequence of second-order differential equations.