Critical Point In The P-laplace Systems

In this thesis, the periodic solutions of p-Laplacian systems with suitable conditions are studied by means of the least action principle and minimax theorems in variational methods.Chapter 1 is devoted to the least action principle and minimax theorems and the problems that will be studied are presented.In Chapter 2, some essential definitions and preliminary theorems concerning variational methods are introduced. Some important lemmas are given and proved.In Chapter 3, we present some applications of the least principle on ordinary p-Laplacian systems. We introduce the results obtained by the least action principle and the study of the transformation of the periodic solutions of the ordinary p-Laplacian systems to the solutions of the corresponding Euler equation. We systematically introduce some results obtained by the least action principle or by the combination of the least action principle and some other theorems. And the periodic solutions of the ordinary p-Laplacian systems are studied with the nonlinearity satisfying the subconvex, the p-sublinear or p-linear conditions.Chapter 4 is devoted to the applications of the minimax principle on ordinary p-Laplacian systems : introducing the saddle point theorem and some other minimax methods combined the uniformly convex property in Banach space W_T^1,p and some results are obtained by minimax methods .In this Chapter, the saddle point theorem is mainly used to the studies of periodic solutions of p-Laplacian systems with the nonlinearity satisfying the p-sublinear or p-linear conditions.