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

Abstract

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