带磁场多临界非局部椭圆问题的多解

摘要/Abstract

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摘要：

该文考虑下列带磁场的多临界非局部椭圆方程

(0.1)

{\begin{aligned} (- i \nabla - A (x))^{2} u & = λ | u |^{p - 2} u + \sum_{s = 1}^{k} (\int_{Ω} \frac{| u (y) |^{2_{s}^{*}}}{| x - y |^{N - α_{s}}} d y) | u |^{2_{s}^{*} - 2} u 在 Ω 中, \\ u & = 0 在 \partial Ω 上 \end{aligned}

$\left\{\begin{aligned}(-{\rm i}\nabla-A(x))^2u&=\lambda|u|^{p-2}u+\sum\limits^k_{s=1}\Big(\int_{\Omega}\frac{|u(y)|^{2^*_s}}{|x-y|^{N-\alpha_s}} {\rm d}y\Big)|u|^{2^*_s-2}u\quad \text{在} \Omega \text{中},\\u&=0\quad \text{在} \partial\Omega \text{上}\\\end{aligned}\right.$

多解的存在性, 其中 $\Omega$ 是 $\mathbb{R}^N$ 中带光滑边界的有界区域, $N\geq4$ , $i$ 是虚数单位, $2^*_s=\frac{N+\alpha_s}{N-2}$ , $N$ -4< $\alpha_s$ < $N$ , $s=1,2$ , $\cdots,k$ $(k\geq 2)$ , $\lambda$ >0 并且 $2\leq p$ < $2^*=\frac{2N}{N-2}$ . 假定磁向量位势 $A(x)= (A_1(x), A_2(x), \cdots, A_N(x))$ 取实值并且满足局部 Hölder 连续. 该文利用 Ljusternik-Schnirelman 理论证明了当 $\lambda$ 较小时, 方程 (1.1) 至少有 cat $_\Omega(\Omega)$ 个非平凡解.

关键词: 多临界椭圆问题, 磁位势, Ljusternik-Schnirelman 理论

Abstract:

In this paper, we consider the existence of multiple solutions of the following multi-critical nonlocal elliptic equations with magnetic field

{\begin{aligned} (- i \nabla - A (x))^{2} u & = λ | u |^{p - 2} u + \sum_{s = 1}^{k} (\int_{Ω} \frac{| u (y) |^{2_{s}^{*}}}{| x - y |^{N - α_{s}}} d y) | u |^{2_{s}^{*} - 2} u i n Ω, \\ u & = 0 o n \partial Ω, \end{aligned}

$\left\{\begin{aligned}(-{\rm i}\nabla-A(x))^2u&=\lambda |u|^{p-2}u+\sum\limits^k_{s=1}\Big(\int_{\Omega}\frac{|u(y)|^{2^*_s}}{|x-y|^{N-\alpha_s}} {\rm d}y\Big)|u|^{2^*_s-2}u\quad {\rm in}\quad \Omega,\\u&=0\quad {\rm on}\quad \partial\Omega,\\\end{aligned}\right.$

where $\Omega$ is bounded domain with smooth boundary in $\mathbb{R}^N$ , $N\geq4$ , i is imaginary unit, $2^*_s=\frac{N+\alpha_s}{N-2}$ with $N-4$ < $\alpha_s$ < $N, s=1,2,\cdots,k$ $(k\geq2)$ , $\lambda$ >0 and $2\leq p$ < $2^*=\frac{2N}{N-2}$ . Suppose the magnetic vector potential $A(x)= (A_1(x), A_2(x),\cdots, A_N(x))$ is real and local Hölder continuous, we show by the Ljusternik-Schnirelman theory that our problem has at least ${\rm cat}_\Omega(\Omega)$ nontrivial solutions for $\lambda$ small.

Key words: Multi-critical elliptic problem, Magnetic potential, Ljusternik-Schnirelman theory

中图分类号:

O175.29

温瑞江, 杨健夫. 带磁场多临界非局部椭圆问题的多解[J]. 数学物理学报, 2024, 44(2): 396-416.

Wen Ruijiang, Yang Jianfu. Multiple Solutions for Multi-Critical Nonlocal Elliptic Problems with Magnetic Field[J]. Acta mathematica scientia,Series A, 2024, 44(2): 396-416.

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