对数薛定谔方程（组）解的存在性集中性和多解性研究

In this project, we would like to investigate the following two types of nonlinear elliptic problems：（1）logarithmic Schrödinger equations；（2）linearly coupled logarithmic Schrödinger systems. The problems mentioned above have important research significance for physics, and are widely used in quantum mechanics, nuclear physics, nonlinear optics, theory of Bose-Einstein condensation and superfluidity. Different to the semilinear Schrödinger equations, both of these two kinds of problems contain logarithmic nonlinear terms, which lead to loss of C^1-smoothness to the corresponding energy functional. Thus, the classical variational methods and critical point theory fail in the study of the problems above. We want to use the variational method and non-smooth critical theory to study the existence and concentration of sign-changing solutions, the existence and multiplicity of positive solutions to logarithmic Schrödinger equations and logarithmic Schrödinger systems respectively. We hoped that through the study of the above problems, explore and propose a new method to handle the elliptic equation with its energy functional without C^1-smoothness, and thus to improve the development of the nonlinear functional analysis. At the same time, we also hoped that this study can promote the development of other subjects.

本项目拟研究下列两类非线性椭圆问题:（1）对数薛定谔方程；（2）线性耦合的对数薛定谔方程组。上述两类问题对物理学具有重要的研究意义，在量子力学，核能物理，非线性光学，玻色-爱因斯坦凝聚态现象以及超流体理论等方面有着广泛的应用。与经典的半线性薛定谔方程（组）不同的是，这两类问题中均含有对数非线性项，方程（组）对应能量泛函不满足C^1光滑性，进而经典的变分方法和临界点理论在研究对数薛定谔方程（组）时失效。我们拟结合变分方法与非光滑分析方法研究上述问题的变号解的存在性和集中性，正解的存在性和多重性。我们希望通过对上述问题的研究，探索并且提出一套针对能量泛函非C^1 的椭圆方程问题的行之有效的研究框架，从而促使非线性泛函分析理论的发展。同时，也希望此项研究能促进其他学科的发展。

本项目主要是在非线性泛函分析框架下，对几类非线性椭圆型偏微分方程（组）中的一些问题进行深入的研究。主要内容包括：（一）应用方向导数与约束极小方法研究了一类含积分-微分算子的对数薛定谔方程正解和变号解的存在性及其渐近行为；（二）探究了位势函数符号以及衰减性质对分数阶薛定谔方程、非线性薛定谔方程组、拟线性薛定谔方程、分数阶Choquard方程、Kirchhoff-Choquard型方程以及Hartree型方程组高能量解的存在性与多重性产生的影响。通过对以上问题的研究，我们深入挖掘了数学与物理的相互联系，并进一步发展了非线性泛函分析中的新方法。