次临界 Choquard 方程的多解

摘要/Abstract

摘要：

该文考虑次临界 Choquard 方程

$\begin{cases} -{\Delta}{u}+(\lambda V(x)+1)u=\Big(\int_{\mathbb{R}^N} \frac{|u(y)|^{p_{\varepsilon}}}{|x-y|^\mu} {\rm d}y\Big)|u|^{p_{\varepsilon}-2}u,\quad x\in\mathbb{R}^N,\\{u\in{{H}}^{1}{(\mathbb{R}^N)}} \end{cases}$(0.1)

多解的存在性, 其中$N>3$,$\lambda$是正实参数,$p_{\varepsilon}=2^\ast_{\mu}-\varepsilon$,$\varepsilon>0$,$0<\mu<N$,$2^\ast_{\mu}=\frac{2N-\mu}{N-2}$是 Hardy-Littlewood-Sobolev 不等式意义下的临界指数. 假定$\Omega:={\rm int}\,V^{-1}(0)$是$\mathbb{R}^N$中非空带光滑边界的有界区域, 利用 Lusternik-Schnirelman 定理,该文证明了当$\lambda$足够大及$\varepsilon$充分小时, 方程(0.1)至少有$cat_\Omega(\Omega)$个正解.

关键词: 次临界 Choquard 方程, Lusternik-Schnirelman 定理, 解的多重性

Abstract:

In this paper, we are concerned with the multiplicity of solutions for the following subcritical Choquard equation

$\begin{equation*} \begin{cases} -{\Delta}{u}+(\lambda V(x)+1)u=\Big(\int_{\mathbb{R}^N}\frac{|u(y)|^{p_{\varepsilon}}}{|x-y|^\mu}{\rm d}y\Big)|u|^{p_{\varepsilon}-2}u,\quad x\in\mathbb{R}^N,\\{u\in{{H}}^{1}{(\mathbb{R}^N)}}, \end{cases}\end{equation*}$

where$N>3$,$\lambda$is a real parameter,$p_{\varepsilon}=2^\ast_{\mu}-\varepsilon$,$\varepsilon>0$,$\mu\in(0,N)$and$2^\ast_{\mu}=\frac{2N-\mu}{N-2}$is the critical Hardy-Littlewood-Sobolev exponent. Suppose that$\Omega:={\rm int}\,V^{-1}(0)$is a nonempty bounded domain in$\mathbb{R}^N$with smooth boundary, using Lusternik-Schnirelman theory, we prove the problem (0.1) has at least$cat_\Omega(\Omega)$positive solutions for$\lambda$large and$\varepsilon$small enough.

Key words: Subcritical Choquard equation, Lusternik-Schnirelman theory, Multiplicity of solutions

中图分类号:

O175.25

温瑞江, 刘范琴, 徐子怡. 次临界 Choquard 方程的多解[J]. 数学物理学报, 2024, 44(1): 60-79.

Wen Ruijiang, Liu Fanqin, Xu Ziyi. Multiplicity of Positive Solutions to Subcritical Choquard Equation[J]. Acta mathematica scientia,Series A, 2024, 44(1): 60-79.

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