Acta mathematica scientia, Series B >

2024 , Vol. 44 >Issue 4: 1373 - 1393

DOI: https://doi.org/10.1007/s10473-024-0411-9

MULTIPLE SOLUTIONS TO CRITICAL MAGNETIC SCHRÖDINGER EQUATIONS

  • Ruijiang WEN ,
  • Jianfu YANG
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  • School of Mathematics and Statistics, Jiangxi Normal University, Nanchang 330022, China
E-mail: jfyang200749@sina.com

Received date: 2023-03-02

  Revised date: 2023-08-08

  Online published: 2024-08-30

Supported by

J. Yang's research was supported by the National Natural Science Foundation of China (12171212).

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Abstract

In this paper, we are concerned with the existence of multiple solutions to the critical magnetic Schrödinger equation

$\begin{matrix}(-{\rm i}\nabla-A(x))^2u+\lambda V(x)u=\mu |u|^{p-2}u+\Big(\int_{\mathbb{R^N}}\frac{|u(y)|^{2^*_\alpha}}{|x-y|^\alpha}{\rm d}y\Big)|u|^{2^*_\alpha-2}u\quad {\rm in}\ \mathbb{R}^N,\end{matrix}$ (0.1)

where $N\geq4$, $2\leq p<2^*$, $2^*_\alpha=\frac{2N-\alpha}{N-2}$ with $0<\alpha<4$, $\lambda>0$, $\mu\in\mathbb{R}$, $A(x)= (A_1(x), A_2(x),\cdots , A_N(x))$ is a real local Hölder continuous vector function, $i$ is the imaginary unit, and $V(x)$ is a real valued potential function on $\mathbb{R}^N$.Supposing that $\Omega={\rm int}\,V^{-1}(0)\subset\mathbb{R}^N$ is bounded, we show that problem (0.1) possesses at least cat$_\Omega(\Omega)$ nontrivial solutions if $\lambda$ is large.

Cite this article

Ruijiang WEN , Jianfu YANG . MULTIPLE SOLUTIONS TO CRITICAL MAGNETIC SCHRÖDINGER EQUATIONS[J]. Acta mathematica scientia, Series B, 2024 , 44(4) : 1373 -1393 . DOI: 10.1007/s10473-024-0411-9

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