Normalized ground states for fractional Kirchhoff equations with critical or supercritical nonlinearity

Abstract

<abstract><p>The aim of this paper is to study the existence of ground states for a class of fractional Kirchhoff type equations with critical or supercritical nonlinearity</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ (a+b\int_{\mathbb{R}^{3}}|(-\bigtriangleup)^{\frac{s}{2}}u|^{2}dx)(-\bigtriangleup)^{s}u = \lambda u +|u|^{q-2 }u+\mu|u|^{p-2}u, \ x\in\mathbb{R}^{3}, $\end{document} </tex-math></disp-formula></p> <p>with prescribed $ L^{2} $-norm mass</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \int_{\mathbb{R}^{3}}u^{2}dx = c^{2} $\end{document} </tex-math></disp-formula></p> <p>where $ s\in(\frac{3}{4}, \ 1), \ a, b, c &gt; 0, \ \frac{6+8s}{3} &lt; q &lt; 2_{s}^{\ast}, \ p\geq 2^{\ast}_{s}\ (2^{\ast}_{s} = \frac{6}{3-2s}), \ \mu &gt; 0 $ and $ \lambda\in \mathbb{R} $ as a Langrange multiplier. By combining an appropriate truncation argument with Moser iteration method, we prove that the existence of normalized solutions for the above equation when the parameter $ \mu $ is sufficiently small.</p></abstract>