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} - Δ u + (λ V (x) + 1) u = (\int_{R^{N}} \frac{| u (y) |^{p_{ε}}}{| x - y |^{μ}} d y) | u |^{p_{ε} - 2} u, x \in R^{N}, \\ u \in H^{1} (R^{N}) \end{cases}

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

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

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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