非齐次拟线性椭圆方程的泛函分析方法

This project is to do the research for existence,multiplicity and the related properties of solutions for generlized nonhomogeneous quasilinear elliptic equations with nonlinear functional analysis tools and methods (including the variational theory and the topological degree theory,etc),elliptic equations theory (mainly for the Maximum Principle) and the Method of Moving Planes.The equations have physical background and applications,and are of great significance in mathematics theory. They are related with the quasilinear elliptic equations such as p(x)-Laplace, and even have a greater degree of nonlinearity. Under the conditions that semilinear elliptic equations are increasingly perfect, quasilinear elliptic equations theory needs people make further development and innovation. Based on the previous and our past work, this project continues to dedicate to solve the following questions:. Existence multiplicity and centralization of the solutions for nonhomogeneous quasilinear elliptic equations in a bounded domain and the whole space respectively by critical point theory and topological degree theory; symmetry monotonicity and priori estimates of solutions by extremum principle and the Method of Moving Planes;the Caffarelli-Kohn-Nirenberg inequality for singular nonhomogeneous quasilinear operators.

本项目主要应用非线性泛函分析工具及方法（包括变分理论和拓扑度理论等）并结合椭圆方程理论（主要为极值原理）及移动平面法研究更一般的非齐次拟线性椭圆方程解的存在性、多解性及解的相关性质. 此类方程不但具有强烈的物理应用背景,而且在数学理论上具有重要意义. 不仅与拟线性椭圆方程如p(x)-Laplace方程有关,而且与其相比具有更大程度的非线性.在半线性椭圆方程研究日益完善的情况下,拟线性椭圆方程的理论尚不完善,急需人们进一步发展和创新.基于前人和我们过去的工作，本项目继续开展该类方程的探索和研究,致力于解决如下问题:. 利用临界点理论和拓扑度理论分别研究有界区域及全空间中非齐次拟线性椭圆方程解的存在性、多解性及解的集中性;利用极值原理及移动平面法讨论解的对称性、单调性并给出解的先验估计;建立非齐次拟线性算子具有奇异时类似于Caffarelli-Kohn-Nirenberg型不等式.

本项目主要应用非线性泛函分析方法研究更一般的非齐次拟线性椭圆方程解的存在性及多解性，解决了如下问题：.1)有界域上非齐次拟线性椭圆方程次临界情况下解的存在性与多解性。.2)有界域上p(x)-laplace方程在临界情况下解的存在性与多解性。.3)带有奇异项的椭圆方程在临界情况下解的存在性与多解性。.4) 有界域及全空间上，分数阶椭圆方程在临界情况下解的存在性与多解性。