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Multiple Solutions for Multi-Critical Nonlocal Elliptic Problems with Magnetic Field

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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

CLC Number:

O175.29

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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