带权的非线性椭圆问题研究

In this proposal,we plan to investigate the existence and properties of solutions to weighted nonlinear elliptic problems,including nonlinear elliptic problems with the fractional Laplacian,Hardy type and Henon type weighted functions. By transforming nonlinear elliptic problems with the fractional Laplacian to problems with weighted elliptic operator in the cylinder of a higher dimensional space,and using critical point theory and theory in the study of nonlinear elliptic equations,we investigate the existence and multiplicity of solutions,the effect of the geometry and topology of the domain to the existence and the number of solutions,as well as various limiting properties of solutions.By the potential theory, we connect nonlinear elliptic equations(systems) with nonlinear integral equations(systems), and study the regularity,symmetry and uniqueness of solutions of nonlinear ellptic problems by nonlinear integral equations(systems).

本项目研究带权的非线性椭圆问题解的存在性和解的性质，其中包括具有分数次Laplace算子的非线性椭圆问题，含Hardy和Henon型权函数的非线性椭圆问题。通过将分数次Laplace算子的非线性椭圆问题转化为高一维空间柱体上的带权的椭圆算子的非线性问题，我们应用临界点理论，结合非线性椭圆问题的研究方法，研究解和多解的存在性，区域的几何和拓扑对解的存在性和多解的存在性的影响，讨论解的各种渐进性质。利用位势理论，将含分数次Laplace算子的非线性椭圆方程（组）与积分方程（组）相联系，通过积分方程（组）研究解的正则性、对称性、唯一性等。

本项目研究带权的非线性椭圆问题解的存在性和解的性质。项目按计划研究了含Hardy和Henon型权函数的非线性椭圆方程和方程组，具有分数次Laplace算子的非线性椭圆问题的解和多解的存在性，区域的几何和拓扑对解的存在性和多解的存在性的影响，讨论了解的各种渐进性质。研究了与具有分数次Laplace算子的非线性椭圆问题相关的积分方程的解的正则性、对称性、唯一性。构造了薛定谔型方程和方程组的无穷多解。发表相关的学术论文26篇。