拟线性Schrödinger方程和非局部Kirchhoff问题研究

In this project, our aim is to study the quasilinear Schrödinger equations and nonlocal Kirchhoff equations which are widely used in physics. The quasilinear Schrödinger equations have important applications in the study of superfluid film, self-orbit of high energy ultrashort laser, microwave of oscillating plasma and soliton instability in the laser heating process. The nonlocal Kirchhoff equations have strong application background in many fields such as non-Newtonian mechanics, Cosmo physics and elastic theory. We intend to study the existence, multiplicity and concentration of solutions of these two kinds of equations by using the variational method, Moser iteration, perturbation technique, the principle of concentration compactness and the theory of regularity. Firstly, the existence and the behavior of the solutions of the equivalent equations are studied by using the variational method and the critical point theory. Secondly, some consistent estimates are established by using the regularity theory and the existence, multiplicity and concentration of the solutions of the two kinds of equations are obtained by using the approximation method. Then, the potential conditions and nonlocal conditions of the two kinds of equations are compared, and find out the influence of quasilinear term and nonlocal term on the solutions of Schrödinger equations.

本项目拟研究物理学中应用非常广泛的拟线性Schrödinger方程和非局部Kirchhoff方程。拟线性Schrödinger方程在研究超流体薄膜、高能超短激光的自轨道、振荡等离子体微波和激光加热过程中孤子不稳定性等问题中都有重要应用，而非局部Kirchhoff方程在非牛顿力学，宇宙物理和弹性理论等诸多领域都有很强的应用背景。我们拟利用变分方法、Moser迭代、扰动技巧、集中紧性原理和正则性理论研究这两类方程解的存在性、多重性和集中性。首先运用变分方法和临界点理论研究等价方程解的存在性以及解的性态，其次运用正则性理论建立一些一致估计并运用逼近方法获得这两类方程解的存在性、多重性和集中性，然后比较这两类方程的位势条件和非局部条件，分析清楚拟线性项和非局部项对Schrödinger方程解的影响。

欧氏空间上的拟线性Schrödinger方程非局部Kirchhoff型方程在物理学中有重要的应用背景，随着偏微分方程理论及相关数学分支研究的深入，人们逐渐关注这些方程解的性质。这既是相关学科领域发展的客观需要，也是偏微分方程理论研究的内在要求。本项目运用变分方法、Pohozaev等式、集中紧性原理、单调性技巧、扰动方法和指标理论获得了这两类方程解的存在性、多重性和集中性。同时，本项目还分别讨论了带有Sobolev临界指数和Hardy-Littlewood-Sobolev临界指数的Kirchhoff型方程正规化基态解的存在性和多重性。项目达到了预期目标，获得了较好的效果，提升了项目组成员的科研水平，加深了对拟线性Schrödinger方程和非局部Kirchhoff方程及相关问题的认识，为后续进一步研究打下了良好的基础。