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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

• 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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