Ground state and nodal solutions for fractional Kirchhoff equation with pure critical growth nonlinearity

Abstract

<p style='text-indent:20px;'>In this paper, we consider the existence of least energy nodal solution and ground state solution, energy doubling property for the following fractional critical problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{cases} -(a+ b\|u\|_{K}^{2})\mathcal{L}_K u+V(x)u = |u|^{2^{\ast}_{\alpha}-2}u+ k f(x,u),&amp;x\in\Omega,\\ u = 0,&amp;x\in\mathbb{R}^{3}\backslash\Omega, \end{cases} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ k $\end{document}</tex-math></inline-formula> is a positive parameter, <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{L}_K $\end{document}</tex-math></inline-formula> stands for a nonlocal fractional operator which is defined with the kernel function <inline-formula><tex-math id="M3">\begin{document}$ K $\end{document}</tex-math></inline-formula>. By using the nodal Nehari manifold method, we obtain a least energy nodal solution <inline-formula><tex-math id="M4">\begin{document}$ u $\end{document}</tex-math></inline-formula> and a ground state solution <inline-formula><tex-math id="M5">\begin{document}$ v $\end{document}</tex-math></inline-formula> to this problem when <inline-formula><tex-math id="M6">\begin{document}$ k\gg1 $\end{document}</tex-math></inline-formula>, where the nonlinear function <inline-formula><tex-math id="M7">\begin{document}$ f:\mathbb{R}^{3}\times\mathbb{R}\rightarrow \mathbb{R} $\end{document}</tex-math></inline-formula> is a Carathéodory function.</p>