Ground states to a Kirchhoff equation with fractional Laplacian

Abstract

<abstract><p>The aim of this paper is to deal with the Kirchhoff type equation involving fractional Laplacian operator</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left(\alpha+\beta \int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{s}{2}}\psi|^{2}\,\mathrm{d} x\right)(-\Delta)^{s}\psi+\kappa \psi = |\psi|^{p-2}\psi \ \ \ \mbox{in} \ \mathbb{R}^{3}, $\end{document} </tex-math></disp-formula></p> <p>where $ \alpha, \beta, \kappa > 0 $ are constants. By constructing a Palais-Smale-Pohozaev sequence at the minimax value $ c_{mp} $, the existence of ground state solutions to this equation for all $ p\in(2, 2_{s}^{*}) $ is established by variational arguments. Furthermore, the decay property of the ground state solution is also investigated.</p></abstract>