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Concentration phenomena for magnetic Kirchhoff equations with critical growth

Chao Ji is partially supported by Shanghai Natural Science Foundation (20ZR1413900, 18ZR1409100). The research of Vicențiu D. Rǎdulescu was supported by a grant of the Romanian Ministry of Research, Innovation and Digitization, CNCS/CCCDI/UEFISCDI, project number PCE 137/2021, within PNCDI Ⅲ. The research of Vicențiu D. Rǎdulescu was also supported by the Slovenian Research Agency program P1-0292.

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  • In this paper, we study the following nonlinear magnetic Kirchhoff equation with critical growth

    $ \begin{align*} \left\{ \begin{aligned} &-\Big(a\epsilon^{2}+b\epsilon\, [u]_{A/\epsilon}^{2}\Big)\Delta_{A/\epsilon} u+V(x)u = f(|u|^{2})u+\vert u\vert^{4}u \quad \hbox{in }\mathbb{R}^3, \\ &u\in H^{1}(\mathbb{R}^{3}, \mathbb{C}), \end{aligned} \right. \end{align*} $

    where $ \epsilon>0 $ is a parameter, $ a, b>0 $ are constants, $ V:\mathbb{R}^{3}\rightarrow \mathbb{R} $ and $ A: \mathbb{R}^{3}\rightarrow \mathbb{R}^{3} $ are continuous potentials, and $ f: \mathbb{R}\rightarrow \mathbb{R} $ is a nonlinear term with subcritical growth. Under a local assumption on the potential $ V $, combining variational methods, penalization techniques and the Ljusternik-Schnirelmann theory, we establish multiplicity and concentration properties of solutions to the above problem for $ \varepsilon $ small. A feature of this paper is that the function $ f $ is assumed to be only continuous, which allows to consider larger classes of nonlinearities in the reaction.

    Mathematics Subject Classification: Primary: 35B38, 35J20, 35J60; Secondary: 49S05, 58E05, 58J90.


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