一类非线性四阶p-Laplace方程的弱解存在性问题

A Class of Weak Solution Existence Problems for Nonlinear Fourth-Order p-Laplace Equations

摘要: 主要介绍了一种证明弱解存在性的一种方法——变分法，变分法的基本内容是确定泛函的极值点和临界点，在一定条件下微分方程边值问题常常可以转化为变分问题来研究。首先通过给定的泛函求极值元，极值点处的方程在分部积分的意义下满足弱解定义，其次构造极小元泛函，将所求问题转化为求解相应泛函的极值元，即得方程弱解的存在性，接下来证明泛函极值元的存在性和弱解的唯一性，从而由变分方法确定该四阶定态p-Laplace方程弱解的存在性问题。

Abstract: This paper introduces a method to prove the existence of weak solutions—variational method. The basic content of variational method is to determine the extreme point and critical point of the functional. Under certain conditions, the boundary value problem of differential equations can often be studied by converting the variational problem. This paper first uses the given functional to find the extreme value element, and the equation at the extreme point satisfies the definition of weak solution in the sense of distribution integral. Secondly, we construct the minimal element functionals, and transform the problem into the corresponding universal extreme element, and we obtain the existence of weak solutions, and next, we prove the uniqueness of weak solutions and the existence of functional extremum elements. we finally give the existence of weak solutions for the weak solutions of the fourth-order stationary p-Laplace equation through the variational method.