量子力学中的非线性偏微分方程

This project mainly focuses on some nonlinear partial differential equations in quantum mechanics by use of the theory and methods of nonlinear functional analysis (including variational theory and topological degree theory etc). These equations not only have intense physical backgrounds and applications but also have important mathematical significance. In particular, these kinds of equations are closely related to the nonlinear fractional elliptic equation, quasilinear higher order elliptic equations and nonlinear Schrödinger equation with the potential function decaying to zero at infinity. However, these equations cannot be treated well so far by use of the standard theory on the partial defferential equastions. Thus, it is required that we should develop this theory further.. This project devotes to the following problems: Using the variational theory and Limit index Theorem to study the existence of solutions and the energy concentration to nonlinear Schrödinger equation with the potential function decaying to zero at infinity; By combining variational methods with topological degree theory to study nonlinear fractional and the quasilinear higher order elliptic equations; Proving the existence, the regularity and the bifurcation of solutions and so on.

本申请项目主要应用非线性泛函的工具和方法(包括变分理论和拓扑度理论等)去研究量子力学中的某些非线性偏微分方程, 这些方程不但具有强烈的物理背景和应用背景, 而且在数学理论上也具有重要的意义. 该类方程与非线性分数阶椭圆方程, 拟线性高阶椭圆型方程和势函数趋于零的非线性Schrödinger 方程密切相关, 而与此相关的偏微分方程理论目前尚不完善, 急需人们进一步发展和创新. 基于前人和我们过去的工作, 我们将继续开展该类非线性偏微分方程的探索和研究.本项目将致力解决如下问题:用变分理论和极限指标定理等研究势函数趋于零的非线性Schrödinger方程解的存在性和解的能量集中问题, 用变分方法结合拓扑度理论研究非线性分数阶和拟线性高阶椭圆方程解的存在性和正则性及解的分歧性质.

本项目应用非线性分析的理论和方法研究量子力学中的某些非线性偏微分方程,这些方程不但具有强烈的物理背景和应用背景,而且在数学理论上也具有重要的意义。该类方程与非线性分数阶椭圆方程,拟线性高阶椭圆型方程和势函数趋于零的非线性 Schrödinger 方程密切相关。我们主要致力解决如下问题: 通过变分理论和极限指标定理等研究势函数趋于零的非线性分数阶 Schrödinger 方程解的存在性和解的能量集中问题,用上同调指标定义特征值的方法结合临界点定理研究非线性分数阶方程,拟线性高阶椭圆方程和具临界非线性项的拟线性椭圆方程,证明解的存在性,多重性,正则性及解的分歧性质等。