非线性椭圆型方程解的多重性和集中性的研究

In this project we mainly use the methods of Nehari manifold and Pohozaev manifold, the global compactness lemma and category theory to study the multiplicity and concentration of solutions for elliptic equations. This research mainly involves the generalized Choquard equation, Kirchhoff problem and fractional Schrödinger equation. These models mainly come from some phenomena in physics and other disciplines, and have important theoretical background and application value. However, the lack of compactness conditions and the appearance of nonlocal operator and integral term make it difficult to study these models. We will seek to break through these difficulties. First of all, we consider the multiplicity and concentration of solutions for generalized Choquard equation with critical growth. Then, when the potential has global minimum or maximum points, and the nonlinearity is concave-convex, we focus on the multiplicity and concentration of solutions for Kirchhoff problem and fractional Schrödinger equation. Concerning Berestycki-Lions type nonlinearity, we pay attention to the multiplicity and concentration of solutions for Choquard equation and fractional Schrödinger equation when the potential has local minimum or maximum points. In addition, regarding to the multiplicity of solutions, we mainly investigate the relation between the number of solutions and the topology of the set where the potential attains its global or local extremum.

本项目利用Nehari流形方法、Pohozaev流形方法、全局紧性引理以及畴数理论等变分方法研究椭圆型方程解的多重性和集中性等问题。 主要涉及广义Choquard方程、Kirchhoff 问题和分数阶Schrödinger方程。这些模型来自于物理学等学科，具有重要的理论背景和应用价值。但是，紧性条件的缺失和非局部算子、积分项的出现使得这些模型的研究有一定的难度，本项目力求突破这些难点。首先，研究临界增长的广义Choquard方程解的多重性和集中性。然后，考虑位势具有全局最值点时凹凸组合的Kirchhoff问题和分数阶Schrödinger方程解的多重性和集中性。另外，研究位势具有局部极值点时带有Berestycki-Lions型非线性项的广义Choquard方程和分数阶Schrödinger方程解的多重性和集中性。 对于解的多重性，着重分析解的个数与位势全局或局部极值点集的拓扑之间的关系。

本项目利用Nehari流形方法、全局紧性引理、畴数理论和罚断方法等研究椭圆型方程解的多重性和集中性等问题。主要涉及广义Choquard方程、Kirchhoff问题、分数阶Schrödinger方程、拟线性Schrödinger方程和Schrödinger-Poisson系统。这些模型来自于物理学等学科，具有重要的理论背景和应用价值。本项目的主要内容有：首先，证明了临界增长的Choquard方程和分数阶Choquard方程解的多重性和集中性。然后，考虑了位势具有全局最值点的三次增长的拟线性Schrödinger方程和Schrödinger-Poisson系统解的存在性和集中性, 位势具有局部极值点的Kirchhoff问题解的多重性和集中性。另外，研究了位势具有临界频率的Schrödinger-Poisson系统和分数阶Schrödinger方程解的多重性。