Solutions to a class of nonlinear Schrödinger equations involving a nonlocal term.

项目研究的方程出现在物理和化学中，与所谓的密度泛函理论有关。所谓密度泛函理论（DFT）的主要思想是用单一粒子框架描述复杂的多体效应。这在数学上转化为使用单个而不是许多强耦合方程，在粒子弱相互作用（例如药物-蛋白质和蛋白质-蛋白质相互作用）的情况下产生准确的描述。这可以解释分子的几何形状和解离能，是解决科学和技术问题的关键，例如理解和设计酶的催化过程，医学上的药物设计等。正如最近的论文所指出的，DFT在处理强相互作用的粒子系统时出现了主要的挑战和不准确性，例如多电子原子。这主要是由于所使用的计算方法无法捕捉无界域上的非线性方程的数学特征。提出的研究旨在集中于一类这样的方程，这些方程是作为描述强相互作用粒子系统（例如原子的电子）的偏微分方程系统（Hartree-Fock方程）的“平均场近似”出现的。在这种情况下，DFT产生一个称为非线性Schrödinger-Poisson-Slater方程的单一方程。拟议研究的主要目的有两个。项目的第一部分是确定条件，以证明涉及函数f的一类方程的无穷多个解的存在性。该函数的存在使所研究的方程更一般，实际上可以解释为文献中可用方程的“电荷校正器”。在第一部分中，为了严格证明所提方程的无穷多个解的存在性，将对一类函数F进行识别。在后期阶段，该项目的目标是实施早期在发现的解决方案（以“能量”为特征）上开发的新知识，以便根据理论的新结果改进现有的计算方法。这需要对现有的计算方法进行重大改进，因为方程是在无界域上设置的。项目第一部分将采用的方法与偏微分方程的非线性分析和变分方法有关。第二部分是计算性的。学生进阶任务目标1：回顾与寻找Schrödinger-Poisson类型系统的多个解相关的数学文献，包括参考文献[5,6]。巩固对抽象的Lusternik-Schnirelman理论的知识，从1990年的第9章和第10章开始，熟悉该理论主要结果的严格证明。目标2：准备关于非线性薛定谔-泊松系统的多重解的存在性的论文初稿，该系统涉及在[3]中推测的非径向电荷密度。目标3：学习简单椭圆型偏微分方程的有限元技术，巩固相关泛函空间知识。主要参考文献：[2,4]。这将更系统地涉及到第二位导师M. Edwards教授。目标4：实现先前开发的理论，以近似计算在目标1和目标2中找到的解决方案。主要参考文献：b[1] A. Ambrosetti和A. Malchiodi。非线性分析与半线性椭圆问题[j]S. Brenner和L. Scott。的数学

Background The equations studied in project arise in Physics and Chemistry in relation to the so-called Density Functional Theory. The main idea of the so-called Density Functional Theory (DFT), consists in describing complex many-body effects with a single particle framework. This mathematically translates into working with a single instead of many strongly coupled equations, yielding an accurate description in instances where particles are weakly interacting (e.g. drug-protein and protein-protein interactions). This allows to explain the geometries and dissociation energies of molecules and is key to tackle problems in science and technology, such as understanding and design of catalytic processes in enzymes, drug design in medicine, etc.. As pointed outin recent papers major challenges and inaccuracies in DFT arise when dealing with strongly interacting particle systems, e.g. multi-electron atoms. This is mainly due to the failure of the used computational methods in capturing the mathematical features of nonlinear equations set on unbounded domains. The proposed research aims at focussing on a class of these equations which arise as "mean field approximations" of systems of partial differential equations (Hartree-Fock equations) describing strongly interacting systems of particles, for instance the electrons of an atom. In this context DFT yields a single equation known as Nonlinear Schrödinger-Poisson-Slater equation. AimsThe main aim of the proposed research is twofold. A first part of the project is to identify conditions in order to prove the existence of infinitely many solutions to a class of these equations which involve a function, say F. The presence of this function makes the equation under study more general, and in fact can be interpreted as a "charge-corrector" to equations available in the literature. In this first part a class of functions F will be identified in order to rigorously prove the existence of infinitely many solutions to the proposed equation. At a later stage the project aim is to implement the new knowledge developed earlier on the solutions found (characterised in terms of "energy"), in order to improve existing computational methods, informed by the new results of the theory. This requires significant improvement of the existing computational methods because the equations are set on unbounded domains.The methodologies which will be followed in the first part of the project are related to the Nonlinear Analysis of Partial Differential Equations and Variational Methods. The second part will be computational. The student progressive tasksTarget 1: To review the mathematical literature related to finding multiple solutions to systems of Schrödinger-Poisson type, including references [5, 6]. To consolidate knowledge of the abstract Lusternik-Schnirelman theory, from [1] chapters 9 and 10 and Brower's degree theory, familiarising with the rigorous proofs of the main results of the theory.Target 2: Preparation of a first draft of the thesis on the existence of multiple solutions to a nonlinear Schrodinger-Poisson system involving a non-radial charge density conjectured in [3].Target 3: To learn the Finite Element Method techniques applied to simple elliptic partial differential equations, and consolidate knowledge in the relevant functional spaces. Main references: [2,4].This will involve more systematically the second supervisor Prof. M. Edwards.Target 4: Implementation of the theory developed earlier to approximate computationally the solutions found within Target 1 and Target 2.Main references:[1] A. Ambrosetti and A. Malchiodi. Nonlinear Analysis and Semilinear Elliptic Problems.[2] S. Brenner and L. Scott. The Mathematic