分数阶Choquard方程正解的存在性、多重性和集中现象

摘要/Abstract

摘要：

该文考虑了下面的次临界的分数阶Choquard方程 $\begin{eqnarray*} \varepsilon^{2s}(-\Delta)^{s}u+V(x)u=\varepsilon^{\mu-N}(|x|^{-\mu}\ast F(u))f(u), \quad x\in\mathbb{R} ^{N} \end{eqnarray*}$ 正解的存在性、多重性和集中现象, 这里 $\varepsilon>0$ 是一个常数, $s\in (0, 1)$ , $(-\Delta)^{s}$ 是分数阶Laplace算子, 位势 $V:\mathbb{R} ^{N}\rightarrow\mathbb{R}$ 是正的且有全局极小, $0<\mu<\min\{4s, N\}$ , 非线性项 $f\in C^{1}(\mathbb{R} , \mathbb{R} )$ 是次临界增长的, $F$ 是 $f$ 的原函数.该文的主要研究方法是变分法和Ljusternik-Schnirelmann理论.

关键词: 分数阶Choquard方程, 变分法, Ljusternik-Schnirelmann理论, 正解, 集中现象

Abstract:

We are concerned with the existence, multiplicity and concentration of positive solutions for the following fractional Choquard equation with subcritical nonlinearity $\begin{eqnarray*} \varepsilon^{2s}(-\Delta)^{s}u+V(x)u=\varepsilon^{\mu-N}(|x|^{-\mu}\ast F(u))f(u), \quad x\in \mathbb{R} ^{N}, \end{eqnarray*}$ where $\varepsilon>0$ is a parameter, $s\in(0, 1)$ , $(-\Delta)^{s}$ is the fractional Laplace operator, $V:\mathbb{R} ^{N}\rightarrow\mathbb{R}$ is a positive potential having global minimum, $0<\mu<\min\{4s, N\}$ , and $F$ is the primitive of $f\in C^{1}(\mathbb{R} , \mathbb{R} )$ which is subcritical growth. The main research methods of this article are variational method and the Ljusternik-Schnirelmann theory.

Key words: Fractional Choquard equation, Variational method, Ljusternik-Schnirelmann theory, Positive solution, Concentrating phenomenon.

中图分类号:

O175.2

张伟强,赵培浩. 分数阶Choquard方程正解的存在性、多重性和集中现象[J]. 数学物理学报, 2022, 42(2): 470-490.

Weiqiang Zhang,Peihao Zhao. Existence, Multiplicity and Concentration of Positive Solutions for a Fractional Choquard Equation[J]. Acta mathematica scientia,Series A, 2022, 42(2): 470-490.

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