两类椭圆型方程解的存在性和多重性的研究

In this project we mainly use the method of Nehari manifold, the concentration compactness lemma and the index or pseudo index theory to study the existence, multiplicity and concentration of solutions for elliptic equations. This research mainly involves the generalized Choquard equation and fractional Schrödinger equation. These models are 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 integral term make it difficult to study these models. We will seek to break through these difficulties. First of all, we consider generalized Choquard equations which are strongly indefinite or with critical growth, and show the existence of ground states and the multiplicity of geometrically distinct solutions. We also look for the multiplicity of positive solutions for generalized Choquard equation with an external source term. Then, under different monotonicity conditions, we focus on the existence of ground states for fractional Schrödinger equation with critical growth. In addition, for fractional Schrödinger equation, we pay attention to the multiplicity of solutions for periodic case and the concentration of solutions for the case that the nonlinearity is indefinite.

本项目主要利用Nehari流形方法、集中紧性引理以及指标理论等变分方法研究椭圆型方程解的存在性、多重性和集中性等问题。 主要涉及广义Choquard方程和分数阶Schrödinger方程。这些模型主要来自于物理学等学科中一些现象，具有重要的理论背景和应用价值。但由于紧性条件的缺失和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，研究强不定或具有临界指数增长的广义Choquard方程基态解的存在性和几何分离解的多重性。考虑外加一干扰项的凹凸组合的广义Choquard方程正解的多重性。然后，在不同的单调性条件下研究具有临界指数增长的分数阶Schrödinger方程基态解的存在性。另外，对于分数阶Schrödinger方程，关注周期情形解的多重性和非线性项不定时解的集中性。

本项目以Nehari流形方法、集中紧性引理以及指标理论等变分方法为主要研究工具，讨论了广义Choquard方程、分数阶Schrödinger方程和Kirchhoff方程等椭圆型方程解的存在性、多重性和集中性等问题。这些问题来自于物理学等学科中一些现象，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，证明了外加一干扰项的凹凸组合的广义Choquard方程正解的多重性。 其次，研究了位势为渐近周期情形时广义Choquard方程基态解的存在性和位势为周期情形时几何分离解的多重性。然后，在不同的单调性条件下，证明了具有临界指数增长的分数阶Schrödinger方程基态解的存在性。另外，在半经典情形下，考虑了位势具有全局最值点时带有三次增长的非线性项的Kirchhoff方程基态解的存在性和集中性。