含非局部项的非线性椭圆型方程组

Nonlinear elliptic systems with nonlocal terms stem from the study of the solitary wave solutions of quantum many-body systems, in which the nonlocal terms reflect the interactions between multiple particles. This project is devoted to the existence and properties of solutions to those systems, especially the ground state solutions and excited state solutions with physical significance. For a nonlinear elliptic system with nonlocal terms which come from the Slater approximation of Hartree-Fork exchange potential, the aims are to prove the existence of the ground state solutions and excited state solutions, to discuss the asymptotic behaviour of the nontrivial solutions as the parameter converge to a point, and to get estimations to the norm of solutions. However, the classical variational methods can't be used in this kind of problems directly, the achivements in this area are few. This project is intended to study the existence of solutions by combining the variational methods and the techniques in partial differential equations, and to study the properties of solutions by using the techniques in partial differential equations such as scaling transformation. The research can help us to learn more about the effect of the nonlocal terms on the existence of solutions of nonlinear elliptic systems, and provide some new methods for studying the nonlinear elliptic systems with nonlocal terms.

含非局部项的非线性椭圆型方程组源于量子多体系统孤立波解的研究，其中的非局部项反映了多个粒子间的相互作用。本项目主要研究它的具有物理意义的解――基态解和激发态解的存在性及相关性质。具体研究内容包括：在Hartree-Fock-Slater理论体系下，证明含非局部项的椭圆型方程组的基态解和激发态解的存在性，并讨论它的非平凡解随参数变化时的渐近行为，给出解的相关范数估计等。目前有关这方面的研究结果较少，经典的变分方法很难直接适用于这类问题。该项目拟用变分方法结合偏微分方程的技巧来研究解的存在性等问题，并考虑应用伸缩变换等预估计方法研究解的性质。我们想通过这些问题的研究更深刻地了解非局部项对椭圆型方程组解的存在性的影响，同时为研究含非局部项的非线性椭圆型方程组提供一些新方法。

本项目研究了含有非局部项的椭圆型方程非平凡解, 多解的存在性以及解的性质。具体内容包括:1.在奇异位势条件下， 利用形变引理和伸缩变换技巧证明了具有凹凸变分泛函结构的Schrödinger Poisson方程非平凡解的存在性，并研究了非平凡解随频率参数趋近于0时的渐近行为。进而获得了 Schrödinger Poisson方程的一类涡旋解。2. 我们研究了含库仑位势的稳态 Schrödinger Poisson Slater 方程, 利用其变分泛函在H^1空间中的下凸性质得到基态解，利用具有库仑位势的Schrödinger算子的特征值研究了激发态解的存在性。3. 通过研究一类含非局部项的椭圆型方程的非平凡解的存在性，我们得到了Chern Simons Schrödinger方程蜗旋解的存在性，多解性和非存在性。