Variational Method Of BVP Of Differential Equations With P-Laplace

In this article, we use variationl methods, especially the dual least action principle and the saddle point theorem to discuss two kinds of differential equations with the p-Laplcian, including the existence of periodic solutions, harmonics and subharmonic ones. The article contains four chapters.The first chapter introduces the background of the differential equations with the -Laplcian, and the main work in this article.The second chapter discusses the periodic boundary value problem of a second-order nonautonomous equation with the p-Laplacian The existence results are obtained of the periodic solutions by using the dual least action principle, perturbation technique and a generalized Wirtinger’s inequality, where the nonlinear term F must increase with p-th growth conditions. This chapter is not only widen the application of the dual least action principle, but also weaken the existing criteria of the periodic solution.The third chapter considers the existence of the periodic solutions and the subhar-monics of the second order differential equations with the p-Laplacian. Applying the dual least principle and the perturbation technique, we get the existence of the kT peri-odic solutions. We further give a sufficient condition for the existence of subharmonics after the priori estimate on all the κT-periodic solutions.The fourth chapter makes a conclusion.