Radial solutions for a fractional Kirchhoff type equation in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^N$$\end{document}

This work is concerned with the following equation of Kirchhoff type involving the fractional Laplacian a+b∫RN(-Δ)s2u2dx(-Δ)su+u=f(u)inRN.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left( a+b\int _{\mathbb {R}^N}\left| (-\varDelta )^{\frac{s}{2}}u\right| ^2dx\right) (-\varDelta )^su+u=f(u) \, \text{ in } \mathbb {R}^N. \end{aligned}$$\end{document}By transforming this equation into an equivalent system, under suitable assumptions we establish the existence of at least two nontrivial radial solutions without (AR) condition. Moreover, the nonexistence of solutions is also investigated.