拟线性椭圆方程解的性质研究

Quasilinear elliptic equations with measure data or Hardy potential, which need more methods and techniques since the solutions cannot be expected to belong to the energy space and non-compact embeddings, have attracted a great deal of famous mathematician’s interests, such as Nirenberg,Caffarelli,Brezis and so on. In this project, we will deal with the existence and qualitative properties of solutions to quasilinear elliptic equation involving measure data or the Hardy potential with the aid of functional analysis, harmonic analysis and theory of partial difference equation.More precisely, firstly, we will concern existence and pointwise estimates of solutions to quasilinear elliptic equations with exponential nonlinearity and measure data, with the help of Wolff’s potentials; Secondly, replacing the locally Lipschitz continuous condition by some technical conditions that involve a suitable function, we obtain comparison principle in narrow domain,furthermore,we will study the monotonicity of solutions to quasilinear elliptic problems by moving plane method; Final,we consider the effect of geometry of boundary and Hardy potential with critical exponent on the existence and monotonicity of solutions by variational principle and moving plane method.This problems is challenging since it is closed related to the best constants in the Caffarelli-Kohn-Nirenberg inequalities.

含测度或Hardy位势的拟线性椭圆方程，由于其解不在能量空间和非紧嵌入，使得该类方程的研究需要新的方法和技巧，引起了诸多数学家如Nirenberg、Caffarelli、Brezis等人的极大兴趣。本项目将综合运用泛函分析，调和分析和偏微分方程理论，研究测度和Hardy位势对拟线性椭圆方程解的存在性和定性性质的影响。具体来说：首先借助Wolff位势，建立含测度和反应项为指数函数的拟线性椭圆方程解的存在性和逐点估计；其次将非线性主部的局部Lipschitz连续性条件转化为具有特殊结构的辅助函数，建立窄带区域的比较原理，进一步应用平面移动法得到一般形式的椭圆方程解的单调性；最后应用变分原理和平面移动法，研究区域的几何性质和含临界指数的Hardy位势对拟线性椭圆方程解的存在性和单调性的影响，该问题与Caffarelli–Kohn–Nirenberg不等式的最优常数密切相关，具有很大的挑战性。

含测度或者Hardy位势的拟线性椭圆方程，由于其解一般不在能量空间和非紧性嵌入，使得该类方程研究需要新的方法和技巧，引起了诸多著名数学家如Nirenberg,Caffarelli、Brezis等人的极大兴趣。本项目将综合应用泛函分析、调和分析和现代偏微分方程理论，研究含测度和含Hardy的拟线性椭圆方程解的存在性和定性性质。具体来说：首先借助Wolff位势和Good-不等式，建立含测度和反应项为指数函数的拟线性方程解的存在性和逐点估计；其次将非线性主部的局部Lipschitz 性条件转化为具有特殊结构辅助函数，建立窄带区域的比较原理，进一步应用平面移动法得到了一般形式的椭圆方程解的单调性，极大地拓展了已有的研究范围；最后应用变分原理和平面移动法发，研究了区域的几何性质和含临界指数的Hardy位势对拟线性方程解的存在性和单调性的影响，该问题与Caffarelli–Kohn–Nirenberg不等式的最优常数密切相关，具有很大的挑战性。