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Ground state and nodal solutions for fractional Schrödinger-Maxwell-Kirchhoff systems with pure critical growth nonlinearity

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The first author is supported by NSF of China (11790271), Guangdong Basic and Applied basic Research Foundation(2020A1515011019), Innovation and Development Project of Guangzhou University

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  • In this paper, we consider the existence of a ground state nodal solution and a ground state solution, energy doubling property and asymptotic behavior of solutions of the following fractional critical problem

    $ \begin{equation*} \begin{cases} (a+ b\int_{\mathbb{R}^{3}}(|(-\Delta)^{\alpha/2}u|^{2})dx)(-\Delta)^{\alpha}u+V(x)u+K(x)\phi u = |u|^{2^{\ast}-2}u+ \kappa f(x,u),\\ (-\Delta)^{\beta}\phi = K(x)u^{2}, \quad x\in\mathbb{R}^{3}, \end{cases} \end{equation*} $

    where $ a, b,\kappa $ are positive parameters, $ \alpha\in(\frac{3}{4},1),\beta\in(0,1) $, and $ 2^{\ast}_{\alpha} = \frac{6}{3-2\alpha} $, $ (-\Delta)^{\alpha} $ stands for the fractional Laplacian. By the nodal Nehari manifold method, for each $ b>0 $, we obtain a ground state nodal solution $ u_{b} $ and a ground-state solution $ v_b $ to this problem when $ \kappa\gg 1 $, where the nonlinear function $ f:\mathbb{R}^{3}\times\mathbb{R}\rightarrow\mathbb{R} $ is a Carathéodory function. We also give an analysis on the behavior of $ u_{b} $ as the parameter $ b\to 0 $.

    Mathematics Subject Classification: Primary: 35J60, 35J50; Secondary: 35Q61.


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