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

Abstract

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 $ (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}, $ with prescribed $ L^{2} $-norm mass $ \int_{\mathbb{R}^{3}}u^{2}dx = c^{2} $ where $ s\in(\frac{3}{4}, \ 1), \ a, b, c > 0, \ \frac{6+8s}{3} < q < 2_{s}^{\ast}, \ p\geq 2^{\ast}_{s}\ (2^{\ast}_{s} = \frac{6}{3-2s}), \ \mu > 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.

Keywords

• normalized solution
• fractional kirchhoff equation
• pohozaev manifold
• moser iteration method
• supercritical growth