非线性椭圆方程组变号解的存在性及相关问题研究

The main purpose of this project is to study the existence of sign-changing solutions of nonlinear elliptic systems by variational method, especially by the method of invariant sets of flows in critical points theory. The method of invariant sets of flows is a powerful tool for seeking sign-changing solutions, but the results are suitable only for a single equation. We need to develop a new critical point theory which is used to prove the existence of sign-changing solutions of nonlinear elliptic systems. Mainly involves three kinds of problems: one is sign-changing solutions for nonlinear Schrödinger systems, the main difficult is to exclude solutions with one or more components being zeroes, and to obtain solutions with each components being sign-changing; another is sign-changing solutions for quasilinear elliptic systems which includes linear growth p-Laplace eauations and modified nonlinear Schrödinger systems, to obtain sign-changing solutions we need add some restraint conditions on nonlinearity. For modified nonlinear Schrödinger systems,we will propose a new approach because this kind of equations have the formal variational structure; and the last is sign-changing solutions for p-biharmonic equations with Hardy potential, the difficulty is that positive and negative part of a function can not be used as the test function, constructing the operator by using the method of invariant sets of flows needs special handling.

本项目拟利用变分方法特别是临界点理论中的流不变方法讨论非线性椭圆方程组变号解的存在性. 流不变方法是寻找变号解的有力工具， 但是已有的结果只适合单个方程. 我们需要寻找新的临界点定理用于证明方程组变号解的存在. 主要涉及三类问题：一类是非线性Schrödinger方程组的变号解, 解决该问题的主要困难在于排除部分分量为零的解，寻找每个分量都变号的解；另一类是拟线性椭圆方程组的变号解，包括线性增长p-Laplace方程组和修正的非线性Schrödinger方程组，要得到变号解需要对非线性项加一些条件，对于修正的非线性Schrödinger方程组，这类方程只有形式上的变分结构，需要寻求新的解决办法； 最后一类是带Hardy项的p-重调和方程组的变号解，困难在于函数的正部和负部不能作为检验函数，用流不变集方法构造算子时要特殊处理.

本项目基本是按选题时所确定的研究内容进行研究的. 在有多个伪梯度流不变集时, 对照经典的山路引理和对称山路引理建立了一般变号临界点定理, 并应用于一类非线性Schrödinger方程组获得无穷多变号解及混合态解的存在性， 发展的新抽象理论不仅可以应用于非线性椭圆方程组，还可用于单个方程获得变号解； 研究了带临界指数的非线性Schrödinger方程组， 先考虑次临界问题， 获得近似解, 然后通过逼近获得了临界情形下无穷多变号解的存在性； 提出了一种新方法--截断方法， 并应用于p-Laplace 方程获得无穷多解的存在性. 新理论和新方法应用于椭圆方程和方程组，取得了一系列有意义的成果，按期完成了项目规定的任务，共发表论文15篇. 本项目的研究成果大多数发表在国际水平的杂志上.